ultra-tight gps/reduced imu for land vehicle navigation · ultra-tight gps/reduced imu for land...

UCGE Reports Number 20305

Department of Geomatics Engineering

Ultra-Tight GPS/Reduced IMU for

Land Vehicle Navigation

(URL: http://www.geomatics.ucalgary.ca/graduatetheses)

By

Debo Sun

March 2010

UNIVERSITY OF CALGARY

Ultra-Tight GPS/Reduced IMU for Land Vehicle Navigation

by

Debo Sun

A THESIS

SUBMITTED TO THE FACULTY OF GRADUATE STUDIES

IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE

DEGREE OF DOCTOR OF PHILOSOPHY

DEPARTMENT OF GEOMATICS ENGINEERING

CALGARY, ALBERTA

MARCH 2010

© Debo Sun 2010

iii

Abstract

The navigation system of a vehicle plays an important role in the vehicle’s safety and

control. Such a system can be realized through the integration of a Global Positioning

System (GPS) receiver and an inertial measurement unit (IMU) to provide more accurate

navigation information than either system alone. To reduce the cost and volume of such

systems, a reduced IMU (RIMU) consisting of only one vertical gyro and two or three

accelerometers is used to integrate with a GPS receiver, resulting in three types of

GPS/RIMU integration strategies, namely loose, tight and ultra-tight (UT). When the

phase lock loops (PLLs) of the GPS receiver in a tight system are aided with the Doppler

shift from the integrated system, the tight system is termed as tight integration with loop

aiding (TLA).

In this dissertation, the RIMU mechanization, TLA GPS/RIMU, and UT GPS/RIMU are

thoroughly researched and evaluated with field vehicle test data. The novel elements of

the work include (i) an innovative local terrain predictor (LTP) algorithm for RIMU, (ii)

an innovative adaptive loop filter (ALF) for TLA GPS/RIMU, and (iii) two innovative

algorithms – namely a cascaded PLL plus a frequency discriminator (CaPF) composite

loop and a reconfigurable tracking loop (RTL) – for UT GPS/RIMU. Furthermore, two

kinds of UT systems – a vector delay lock loop plus a vector frequency lock loop (VDF)

and a vector delay lock loop plus a cascaded PLL (VDCaP) – are implemented.

Test results show that all of the above innovative algorithms are valid, and the three types

of GPS/RIMU (i.e. loose, TLA, and UT) work well. Specifically, the LTP method can

iv

reduce the three dimensional (3D) root mean square (RMS) velocity error by more than

80% compared to without LTP case; the ALF algorithm can reduce the 3D RMS velocity

error by up to 19% compared to constant noise bandwidth loop filters; the VDF and

VDCaP systems work well in land vehicle navigation, and the CaPF and RTL algorithms

can potentially improve the navigation performance of the UT system.

v

Acknowledgements

I would like to express my sincere gratitude to my supervisor, Dr. M. Elizabeth Cannon,

for her invaluable support, guidance and encouragement throughout my studies and

research. I really appreciate that she gave me the opportunity to study in the Position,

Location And Navigation (PLAN) group.

I would like to extend my heartfelt gratitude to my co-supervisor, Dr. Mark G. Petovello,

for his valuable helps and advices. I am also highly indebted to Dr. Gérard Lachapelle.

As my first co-supervisor, he guided my studies and research in my first one and half

years and helped me in many ways.

I would further like to extend my sincere thanks to Dr. Cillian O'Driscoll, Jared Bancroft,

Tao Lin, and other members and ex-members of the PLAN group for their help in various

ways. My special thanks must go to Dr. Guojiang Gao, Dr. Jiancheng Gao, Dr. Ali

Broumandan, Dr. Florence Macchi, Dr. Cyrille Gernot, Aiden Morrison, Tao Li, Richard

Ong, Kannan Muthuraman, Vahid Dehghanian, Pejman Kazemi, Peng Xie, Wang Da,

Ping Luo, Wei Cao, for graciously sharing their time with me and their support.

Most of all, I thank my family for their love and greatest support, and I would like to

dedicate this dissertation to my wife Lisa and my lovely daughter Jennifer.

vi

Table of Contents

Approval Page..................................................................................................................... ii

Abstract .............................................................................................................................. iii

Acknowledgements..............................................................................................................v

Table of Contents............................................................................................................... vi

List of Tables ...................................................................................................................... x

List of Figures ................................................................................................................... xii

List of Symbols, Abbreviations and Nomenclature......................................................... xvi

CHAPTER ONE: INTRODUCTION..................................................................................1

1.1 Background................................................................................................................1

1.1.1 Inertial Measurement Unit.................................................................................1

1.1.2 GPS Receiver.....................................................................................................2

1.1.3 GPS/IMU Integration Strategies........................................................................4

1.1.4 Navigation Filter................................................................................................5

1.2 Overview and Limitations of Previous Work ............................................................6

1.2.1 Reduced IMU ....................................................................................................7

1.2.2 GPS Receiver.....................................................................................................9

1.2.3 TLA GPS/Reduced IMU ...................................................................................9

1.2.4 UT GPS/Reduced IMU....................................................................................11

1.3 Research Objectives and Contributions...................................................................14

1.4 Dissertation Outline .................................................................................................19

CHAPTER TWO: OVERVIEW OF LAND VEHICLE NAVIGATION SYSTEMS ......22

2.1 Reference Frames ....................................................................................................23

2.2 Inertial Navigation System ......................................................................................24

2.2.1 Full IMU System .............................................................................................25

2.2.1.1 Navigation Equations.............................................................................25

2.2.1.2 Error Model............................................................................................28

2.2.2 Reduced IMU System......................................................................................31

2.3 GPS Receiver ...........................................................................................................33

2.3.1 GPS Receiver Configuration ...........................................................................33

2.3.2 GPS Receiver Scalar-Based Tracking Loops ..................................................35

2.3.3 GPS Receiver Vector-Based Tracking Loops .................................................38

2.3.4 GPS Receiver Measurements ..........................................................................40

2.3.5 Estimation of Position, Velocity, and Time ....................................................42

2.4 GPS/IMU Integration System..................................................................................44

2.4.1 Loose Integration System ................................................................................44

2.4.2 Tight and Ultra-Tight Integration Systems......................................................46

CHAPTER THREE: MECHANIZATION OF REDUCED IMU.....................................50

3.1 Mechanization of RIMU with a Local Terrain Predictor ........................................50

3.1.1 Derivation of Navigation Equations of RIMU ................................................51

3.1.2 Derivation of Error Model of RIMU ...............................................................54

3.1.3 Equation Summary for 3A1G Configuration ..................................................57

3.1.4 Mechanization and Error Model of 2A1G Configuration ...............................58

vii

3.2 Comparison of Three Types of RIMU Mechanizations and Corresponding

Error Models ..........................................................................................................60

3.2.1 DR Mechanization and Corresponding Error Model ......................................61

3.2.2 FD Mechanization and Corresponding Error Model.......................................63

3.2.3 Comparison of Three Types of Mechanizations and Corresponding Error

Models..............................................................................................................64

3.3 Field Vehicle Test ....................................................................................................67

3.3.1 Test Setup, Route Selection, and Data Collection...........................................67

3.3.2 GPS Availability, Test Trajectory, and Reference Velocity and Attitude.......71

3.4 Data Processing........................................................................................................74

3.5 Evaluation of GPS/RIMU with an LTP...................................................................75

3.5.1 GPS/3A1G Integration ....................................................................................76

3.5.2 GPS/2A1G Integration ....................................................................................78

3.5.3 GPS Outage Test .............................................................................................81

3.5.4 Summary..........................................................................................................85

3.6 Evaluation of Three Types of RIMU M&E Equations............................................86

3.6.1 Test Results with Section A (Near Open Sky Case) .......................................86

3.6.2 Test Results with Section B (Near Foliage Case) ...........................................88

3.6.3 Summary and Discussion ................................................................................90

3.7 Summary..................................................................................................................91

CHAPTER FOUR: TLA GPS/REDUCED IMU...............................................................93

4.1 TLA GPS/RIMU......................................................................................................94

4.2 Phase Error of Doppler-Aided PLL in TLA GPS/RIMU ........................................96

4.2.1 Individual Phase Errors in Doppler-Aided PLL..............................................96

4.2.2 Total Phase Error in Doppler-Aided PLL .......................................................98

4.3 Adaptive PLL Loop Filter for TLA GPS/RIMU .....................................................99

4.3.1 Evaluation of Doppler Aiding Uncertainty for TLA GPS/RIMU .................101

4.3.2 Adaptive PLL Loop Filter Design.................................................................105

4.4 Test Description.....................................................................................................107

4.4.1 Translation Stage Test ...................................................................................108

4.4.2 Field Vehicle Test..........................................................................................110

4.4.2.1 Open Sky Test......................................................................................111

4.4.2.2 Foliage Test..........................................................................................112

4.5 Data Processing and Test Scenario ........................................................................115

4.6 Evaluation of TLA GPS/RIMU .............................................................................117

4.6.1 Test Results with Translation Stage Test Data..............................................118

4.6.2 Test Results with Vehicle Test Data .............................................................120

4.6.3 Summary........................................................................................................123

4.7 Evaluation of TLA GPS/RIMU with ALFs...........................................................124

4.7.1 Test Results with Open Sky Test Data ..........................................................125

4.7.1.1 Velocity and Attitude Error .................................................................125

4.7.1.2 PLL Tracking Performance Comparison between ALF and CNBLF .126

4.7.1.3 Navigation Performance Comparison between ALF and CNBLF ......129

4.7.2 Test Results with Foliage Test Data ..............................................................132

4.7.2.1 PLL Tracking Performance Comparison between ALF and CNBLF .132

4.7.2.2 Navigation Performance Comparison between ALF and CNBLF ......135

viii

4.7.3 Summary and Discussion ..............................................................................136

4.8 Summary................................................................................................................138

CHAPTER FIVE: ULTRA-TIGHT GPS/REDUCED IMU ...........................................140

5.1 UT GPS/RIMU ......................................................................................................141

5.1.1 UT GPS/RIMU with VDF.............................................................................142

5.1.1.1 VDF Implementation ...........................................................................142

5.1.1.2 VDF Measurement Noise ....................................................................147

5.1.2 UT GPS/RIMU with VDCaP ........................................................................149

5.1.2.1 VDCaP Implementation.......................................................................150

5.1.2.2 Doppler Measurement Noise ...............................................................155

5.2 CaPF Composite Loop for UT GPS/RIMU...........................................................158

5.3 Reconfigurable Tracking Loops for UT GPS/RIMU ............................................164

5.3.1 Switch Strategy between VDLL and DLL ....................................................165

5.3.2 Switch Strategy between CaPF and PLL.......................................................167

5.3.3 Implementation of Reconfigurable Tracking Loops (RTLs).........................170

5.4 Theoretical Comparison of UT and TLA GPS/RIMU...........................................171

5.4.1 Comparison in Carrier/Code Tracking Loops ...............................................172

5.4.2 Comparison in System Performance .............................................................174

5.5 Test Description and Data Processing ...................................................................175

5.6 Evaluation of UT GPS/RIMU................................................................................177

5.6.1 Evaluation of GPS/RIMU with VDF ............................................................177

5.6.2 Evaluation of GPS/RIMU with VDCaP ........................................................180

5.6.3 Evaluation of GPS/RIMU with VDCaPF......................................................183

5.6.4 Evaluation of GPS/RIMU with RTLs ...........................................................187

5.6.4.1 Test Results with Open Sky Data ........................................................187

5.6.4.2 Test Results with Foliage Data ............................................................190

5.6.4.3 Summary..............................................................................................192

5.6.5 Summary........................................................................................................192

5.7 Evaluation of UT and TLA GPS/RIMU ................................................................194

5.7.1 Test Results with Open Sky Data..................................................................194

5.7.2 Test Results with Foliage Data......................................................................195

5.7.3 Summary and Discussion ..............................................................................199

5.8 Summary................................................................................................................200

CHAPTER SIX: CONCLUSIONS AND RECOMMENDATIONS ..............................203

6.1 Conclusions............................................................................................................205

6.2 Recommendations..................................................................................................208

REFERENCES ................................................................................................................213

APPENDIX A: MECHANIZATION EQUATIONS AND ERROR MODEL OF

FULL IMU..............................................................................................................228

A.1. Navigation Equations of Full IMU ......................................................................228

A.2. Corresponding Error Model of Full IMU ............................................................229

APPENDIX B: INDIVIDUAL PHASE ERRORS OF DOPPLER-AIDED PLL ...........232

ix

B.1. Thermal Noise and Oscillator Noise-Induced Phase Jitters.................................232

B.2. Doppler Aiding Error-Induced Phase Errors........................................................234

B.2.1. Random Ramp-Induced Phase Error ...........................................................234

B.2.2. First-Order GM-Induced Phase Jitter ..........................................................234

x

List of Tables

Table 3.1 Attitude and Velocity Error Statistics of GPS/3A1G with LTP ....................... 78

Table 3.2 Attitude and Velocity Error Statistics of GPS/2A1G with LTP ....................... 79

Table 3.3 Attitude and Velocity Error Statistics of GPS/RIMU (Crista) without LTP .... 81

Table 3.4 Horizontal and Vertical Position Error Comparison between GPS/RIMU

with and without LTP at The End of GPS Outage.................................................... 85

Table 3.5 Velocity and Attitude Error Statistics of GPS/3A1G (Crista) with Data from

Section A................................................................................................................... 87


Section A................................................................................................................... 88


Section B................................................................................................................... 90


Section B................................................................................................................... 90

Table 4.1 Velocity and Attitude Error of TLA and Tight GPS/3A1G (Crista) with

Open Sky Data ........................................................................................................ 123


Foliage Data ............................................................................................................ 123

Table 4.3 Position Error of TLA GPS/3A1G (Crista) .................................................... 123

Table 4.4 RMS Doppler Errors of Tracked SVs in TLA GPS/3A1G (Crista) with and

without ALFs for Open Sky Test............................................................................ 129



Table 4.6 Velocity and Attitude Error of TLA GPS/RIMU (Crista) with and without

ALFs for Open Sky Test ......................................................................................... 132

Table 4.7 Velocity and Attitude Error of TLA GPS/RIMU (HG1700) with and



without ALFs for Foliage Test................................................................................ 134



xi


ALFs for Foliage Test ............................................................................................. 136



Table 5.1 Velocity and Attitude Error of UT GPS/RIMU (Crista) with VDF and Open

Sky Data.................................................................................................................. 177

Table 5.2 Velocity and Attitude Error of UT GPS/RIMU (Crista) with VDF and

Foliage Data ............................................................................................................ 180

Table 5.3 Velocity and Attitude Error of UT GPS/RIMU (Crista) with VDCaP and

Open Sky Data ........................................................................................................ 181


Foliage Data ............................................................................................................ 182

Table 5.5 Velocity and Attitude Error of UT GPS/RIMU (Crista) with VDCaPF and

Open Sky Data ........................................................................................................ 184


Foliage Data ............................................................................................................ 184


Open Sky Data (with Same Loop Filter Reset Criteria as CaPF) ........................... 184


Foliage Data (with Same Loop Filter Reset Criteria as CaPF) ............................... 184


Open Sky Data When ALF Is Applied ................................................................... 185

Table 5.10 Position, Velocity, and Attitude Error of UT and TLA GPS/RIMU (Crista)

with Open Sky Data ................................................................................................ 195

Table 5.11 Position, Velocity, and Attitude Error of UT and TLA GPS/RIMU (Crista)

with Foliage Data .................................................................................................... 198

Table 5.12 Position, Velocity, and Attitude Error of TLA GPS/2A1G (Crista) With a

DaPLL and Foliage Data ........................................................................................ 198

xii

List of Figures

Figure 2.1 Position Vector in Inertial Frame .................................................................... 26

Figure 2.2 Reduced IMU Configuration........................................................................... 32

Figure 2.3 GPS Receiver Functional Configuration ......................................................... 34

Figure 2.4 Scalar-Based Tracking Loops of GPS Receiver (One Channel) ..................... 36

Figure 2.5 Block Diagram of Linearized Phase Lock Loop (One Channel) .................... 37

Figure 2.6 Vector-Based Tracking Loops of GPS Receiver (One Channel) .................... 39

Figure 2.7 Block Diagram of VDLL’s Lock Loop (One Channel) .................................. 40

Figure 2.8 Loose GPS/INS Integration Block Diagram ................................................... 45

Figure 2.9 Tight GPS/INS Integration Block Diagram .................................................... 47

Figure 2.10 TLA GPS/INS Integration Block Diagram .................................................. 48

Figure 2.11 UT GPS/INS Integration Block Diagram...................................................... 49

Figure 3.1 Vehicle Test Setup........................................................................................... 69

Figure 3.2 Data Collection Block Diagram ...................................................................... 69

Figure 3.3 Route Map of Open Sky Test .......................................................................... 70

Figure 3.4 Route Map of Foliage Test .............................................................................. 70

Figure 3.5 Number of Tracked Satellites in Open Sky Test ............................................. 71

Figure 3.6 Trajectory of Open Sky Test ........................................................................... 73

Figure 3.7 Reference Velocity of Open Sky Test ............................................................. 73

Figure 3.8 Reference Pitch, Roll and Azimuth of Open Sky Test .................................... 74

Figure 3.9 Data Processing Block Diagram...................................................................... 75

Figure 3.10 Velocity and Attitude Errors of GPS/3A1G (HG1700) with LTP ................ 77

Figure 3.11 Velocity and Attitude Errors of GPS/3A1G (Crista) with LTP .................... 78

Figure 3.12 Velocity and Attitude Errors of GPS/2A1G (HG1700) with LTP ................ 79

Figure 3.13 Velocity and Attitude Errors of GPS/2A1G (Crista) with LTP .................... 79

xiii

Figure 3.14 Horizontal Position and Velocity Error Comparison between GPS/RIMU

(Crista) with and without LTP .................................................................................. 84

Figure 3.15 Vertical Position and Velocity Error Comparison between GPS/RIMU

(Crista) with and without LTP .................................................................................. 84

Figure 3.16 Vertical Velocity Error of GPS/RIMU (Crista) with FD Model in Section

B................................................................................................................................ 90

Figure 4.1 Configuration of TLA GPS/RIMU.................................................................. 96

Figure 4.2 Block Diagram of Doppler Aiding Error Propagation .................................... 96

Figure 4.3 Total PLL Phase Error Variation with Noise Bandwidth.............................. 106

Figure 4.4 Noise Bandwidth Search Flowchart .............................................................. 107

Figure 4.5 Translation Stage Test Setup ......................................................................... 109

Figure 4.6 GPS Satellite Skyplot of Translation Stage Test ........................................... 110

Figure 4.7 Reference Velocity of Stage Motion in Forward Direction .......................... 110

Figure 4.8 GPS Satellite Skyplot of Open Sky Test ....................................................... 111

Figure 4.9 Satellite C/N0 Variations of Open Sky Test .................................................. 112

Figure 4.10 Trajectory of Foliage Test ........................................................................... 113

Figure 4.11 Reference Velocity of Foliage Test ............................................................. 113

Figure 4.12 Reference Pitch, Roll and Azimuth of Foliage Test.................................... 114

Figure 4.13 GPS Satellite Skyplot of Foliage Test ......................................................... 115

Figure 4.14 Satellite C/N0 Variations of Foliage Test .................................................... 115

Figure 4.15 Power Curve of Vehicle’s Random Vibration............................................. 117

Figure 4.16 Tracking Performance Comparison of TLA and Tight GPS/2A1G

(HG1700) for PRN19.............................................................................................. 119

Figure 4.17 Forward Velocity Errors of TLA and Tight GPS/2A1G (HG1700)............ 120

Figure 4.18 Number of SVs Used in Navigation Solution of TLA and Tight

GPS/3A1G (Crista) with Open Sky Data................................................................ 122


GPS/3A1G (Crista) with Foliage Data ................................................................... 122

xiv

Figure 4.20 Velocity and Attitude Error of TLA GPS/3A1G (Crista) with ALFs and

Open Sky Data ........................................................................................................ 125


Open Sky Data ........................................................................................................ 126

Figure 4.22 Tracking Performance of TLA GPS/3A1G (Crista) with and without

ALFs for Open Sky Test and PRN 27 .................................................................... 128


ALFs for Open Sky Test and PRN 27 .................................................................... 129


ALFs for Foliage Test and PRN 25 ........................................................................ 134


ALFs for Foliage Test and PRN 25 ........................................................................ 134

Figure 5.1 Vector-Based Tracking Loops of GPS Receiver (One Channel) .................. 143

Figure 5.2 Block Diagram of VDLL’s Lock Loop (One Channel) ................................ 144

Figure 5.3 Block Diagram of VFLL Tracking Loop (One Channel).............................. 145

Figure 5.4 Block Diagram of Cascaded PLL (One Channel) ......................................... 151

Figure 5.5 Block Diagram of Second-Order Loop Filter................................................ 152

Figure 5.6 Block Diagram of First-Order Phase Lock Loop .......................................... 156

Figure 5.7 Doppler Measurement Noise Variances of PLL and VFLL......................... 158

Figure 5.8 Block Diagram of CaPF (One Channel)........................................................ 161

Figure 5.9 Flowchart of Reconfigurable Tracking Loop Implementation...................... 171

Figure 5.10 Velocity and Attitude Error of UT GPS/3A1G (Crista) with VDF and

Foliage Data ............................................................................................................ 179


Foliage Data ............................................................................................................ 179

Figure 5.12 Number of SVs Used in Navigation Solution of UT GPS/RIMU (Crista)

with VDF and Foliage Data .................................................................................... 180


with VDCaP and Open Sky Data............................................................................ 181

xv


with VDCaP and Foliage Data................................................................................ 182

Figure 5.15 Number of SVs Used in Navigation Solution and Vector Tracking Status

of UT GPS/3A1G (Crista) with RTLs and Open Sky Data .................................... 188

Figure 5.16 Velocity and Attitude Error of UT GPS/3A1G (Crista) with RTLs and

Open Sky Data ........................................................................................................ 189


of UT GPS/3A1G (Crista) with RTLs and Foliage Data ........................................ 191


Foliage Data ............................................................................................................ 191


of UT GPS/2A1G (Crista) with RTLs and Foliage Data ........................................ 192

Figure 5.20 Number of SVs Used in Navigation Solution of UT and TLA GPS/3A1G

(Crista) with Foliage Data....................................................................................... 196


(Crista) with Foliage Data....................................................................................... 196

Figure 5.22 Position Error of UT and TLA GPS/3A1G (Crista) with Foliage Data ...... 198

xvi

List of Symbols, Abbreviations and Nomenclature

Symbols Definition

( )( )si Laplace transform of signal ( )i or transfer function

( )Disi Discriminator output of GPS receiver ( )⋅i Time derivative of quantity ( )i

( )i i

i Second time derivative of quantity ( )i

( )i Time average of quantity ( )i

( )i Quantity ( )i with error

( )i Estimate of quantity ( )i

( )∆ i Change in quantity ( )i

( )δ i Perturbation of quantity ( )i

[ ]( ) ×i Skew-symmetric matrix form of vector ( )i

n n×0 Zero matrix with dimension n n×

( )j gA Rotation matrix about j -axis by angle g

( )bi

Accelerometer bias in axis ( )i

b Accelerometer bias vector

nB Noise bandwidth

c Speed of light

0/C N Carrier-to-noise-power density ratio

( )di

Gyro drift in axis ( )i

d Gyro drift vector

D Early-to-late correlator spacing of code discriminator

E Skew-symmetric matrix of attitude error vector ε

df Carrier Doppler shift

daf Carrier Doppler shift aiding

dmf Carrier Doppler shift measurement

LFf Doppler error output of loop filter

Tf Transmitted signal frequency

f Specific force vector

F Skew-symmetric matrix form of vector f g Acceleration of gravity

eg Gravity vector i

G Gravitational acceleration vector

h Geodetic altitude

H Measurement matrix of Kalman filter

I GPS in-phase signal

n n×I Unit matrix with dimension n n×

xvii

L Number of discriminator outputs in a measurement update period

of integration Kalman filter

M Geodetic meridian radius

N Geodetic prime vertical radius

Oρ GPS satellite orbital error

p Vehicle pitch

( )( )−Pi

Prediction error covariance matrix of subscripted quantity of

integration Kalman filter

Q GPS quadraphase signal

r Geometric range between user and GPS satellite

r Vehicle roll ar Vector r in frame a

r Position vector b

aR Rotation matrix from frame a to b

utɺ GPS receiver clock drift rate

T Measurement update period of integration Kalman filter

( )Ti

Threshold of quantity ( )i

utδ GPS receiver clock offset

stδ GPS satellite clock offset

PITT Predetection integration time of GPS receiver

( )vi

Velocity in axis ( )i

v Velocity vector

uv User velocity vector

V GPS satellite velocity vector

( )Var g Variance of quantity g

( )wi

Noise of subscripted quantity

( )Wi

Noise vector of subscripted quantity

X State vector of integration Kalman filter

Z Measurement vector of integration Kalman filter

α Inverse of correlation time of a first-order GM process

ε Attitude error vector

( )εi

Attitude error in axis ( )i

( )εi

Noise of subscripted quantity

κ User-to-satellite line-of-sight unit vector

λ Wavelength of signal

λ Geodetic longitude

ξ Damping ratio of second-order system

iρ GPS pseudorange of i th satellite

( )σi

Standard deviation of subscripted quantity 2

( )σi

Variance of subscripted quantity

xviii

τ code phase of DLL

φ Carrier phase measurement of GPS receiver ϕ Geodetic latitude ϕ Carrier phase of PLL ψ Vehicle azimuth

a

pqΩ Skew-symmetric matrix form of vector a

pqω

( )ωi

Rotation rate in axis ( )i

eω Rotation rate of the earth

nω Undamped natural frequency of tracking loop

a

pqω Vector of rotation rates of frame q, relative to frame p, expressed

in frame a

Abbreviations Definition

2A1G Two horizontal accelerometers plus one vertical gyro

3A1G Three accelerometers plus one vertical gyro

3D Three dimensions

ALF Adaptive loop filter

CaPF Cascaded PLL plus FLL discriminator

CaPLL Cascaded PLL

CNB Constant noise bandwidth

CNBLF Constant noise bandwidth loop filter

CTL Carrier tracking loop

DaDLL Doppler-aided DLL

DaPLL Doppler-aided PLL

DGPS Differential GPS

DLL Delay lock loop

DR Dead reckoning

EKF Extended Kalman filter

FD Full dimension

FLL Frequency lock loop

GM Gaussian Markov

GPS Global positioning system

IF Intermediate frequency

IMU Inertial measurement unit

INS Inertial navigation system

LKF Linearized Kalman filter

LTP Local terrain predictor

LVN Land vehicle navigation

LVNS Land vehicle navigation system

M&E Mechanization and error

MEMS Micro-electro-mechanical system

NCO Numerically controlled oscillator

OCXO Oven controlled crystal oscillator

PIT Predetection integration time

xix

PLI Phase lock indicator

PLL Phase lock loop

RMS Root mean square

PPS Pulse-per-second

PRN Pseudo-random noise

PSD Power spectral density

PVT Position, velocity and time

RIMU Reduced inertial measurement unit

RR Random ramp

RTK Real time kinematic

RTL Reconfigurable tracking loop

SBTL Scalar-based tracking loop

SBTS Scalar-based tracking scheme

SCTL Scalar carrier tracking loop

SV Space vehicle

TCXO Temperature compensated crystal oscillator

TLA Tight integration with loop aiding

UKF Unscented Kalman filter

UT Ultra-tight

VBTL Vector-based tracking loop

VBTS Vector-based tracking scheme

VCTL Vector carrier tracking loop

VDCaP VDLL plus cascaded PLL

VDCaPF VDLL plus CaPF

VDF VDLL plus VFLL

VDLL Vector delay lock loop

VDP VDLL plus VPLL

VFLL Vector frequency lock loop

VPLL Vector phase lock loop

ZUPT Zero velocity update

1

Chapter One: Introduction

With recent automobile technology development, vehicle safety and control have been

given more attention both in civil and military applications. To this end, land vehicle

navigation systems (LVNSs) play an important role because they can provide necessary

information for vehicle navigation and control (Allen et al 2009), and thus make a vehicle

safer, more comfortable, and easier to operate. An LVNS can be obtained through the

integration of a Global Positioning System (GPS) receiver and an inertial measurement

unit (IMU) or with vehicle sensors, to provide more accurate navigation information than

either system alone. In this dissertation, different kinds of GPS/reduced IMU integration

systems such as tight GPS/reduced IMU with Doppler loop aiding and ultra-tight (UT)

GPS/reduced IMU, both of which can be used as an LNVS, are investigated in order to

provide useful information for practical LVNS development.

1.1 Background

GPS/IMU integration systems have been used in many areas, such as flight guidance,

vehicle navigation and geodetic surveying. With continuing technology development,

there exists a growing demand for small, low cost, reliable and high precision GPS

receivers or navigators. The combination of modern GPS and inertial navigation

technology makes this kind of GPS receiver possible. Any GPS/IMU system generally

contains four elements: an IMU, a GPS receiver, an integration strategy, and a navigation

filter. Each of these is discussed briefly below.

1.1.1 Inertial Measurement Unit

An inertial measurement unit is a cluster of instruments (gyroscopes and accelerometers)

that measure angular rates and linear accelerations. Combining an IMU with navigation

2

software, an inertial navigation system (INS) is obtained. An INS is a dead reckoning

system which depends on accelerometer and gyro measurements to calculate changes in

position, velocity and attitude. It is self-contained and can provide accurate position and

velocity over short time periods, but it drifts over time (El-Sheimy 2007; Titterton &

Weston 2004; Farrell & Barth 1999). Generally, an INS consists of three orthogonal

accelerometers and three orthogonal gyros. The gyro measurements are used to calculate

attitude while the accelerometer measurements are used to calculate velocity and position.

In land vehicle navigation, however, in order to reduce the system’s cost and volume, it is

desired to use a minimum set of hardware components or inertial sensors For example,

two horizontal accelerometers and one vertical gyro could be used (Allen et al 2009; Sun

et al 2008; Niu et al 2007b; Syed et al 2007). This kind of IMU with less than three gyros

and/or less than three accelerometers is herein called a “reduced IMU”. For an LVNS, a

reduced IMU (RIMU) usually consists of only one vertical gyro and two or three

accelerometers, which are often substituted with on vehicle sensors such as a yaw rate

sensor and G-sensors (Gao 2007b; Rezaei & Sengupta 2007; Gao et al 2006).

1.1.2 GPS Receiver

In a conventional GPS receiver, generally two kinds of tracking loop are used, namely a

code correlation loop or a delay lock loop (DLL) and a carrier Doppler removal loop

consisting of one or both of a phase lock loop (PLL) and a frequency lock loop (FLL).

The PLL or FLL is for Doppler removal, whereas the DLL is for code correlation. Both

tracking loops work interactively. In order to maintain tracking in high dynamic

environments, both the order and the noise bandwidth of the PLL in a conventional

3

receiver are required to be large (Ward et al 2006), such as for a second order loop with a

noise bandwidth of more than 18 Hz (Babu & Wang 2005). Even in low dynamic

environments, the noise bandwidth is required to be about 5 to 12 Hz for a second order

loop. But if the Doppler can be computed from the inertial solution (and knowledge of

the satellite position and velocity), it can be used to aid the tracking loops, thus allowing

the noise bandwidth of the PLL to be reduced to 3 Hz or less while still tracking in high

dynamic environment (Petovello et al 2007; Babu & Wang 2005). As a result, both the

tracking and the navigation performance of such an inertial-aided receiver can be

improved (Petovello et al 2007).

In order to further improve the receiver’s tracking performance, a vector-based tracking

scheme can be applied both in the code and carrier loops. In the code correlation loop, the

vector-based tracking loop (VBTL) is called vector delay lock loop (VDLL), and in the

carrier Doppler removal loop it is called vector phase lock loop (VPLL) for a PLL and

vector frequency lock loop (VFLL) for an FLL. In the VBTL concept, the phase and

frequency/Doppler (of the code or carrier, as appropriate) of the numerically controlled

oscillator (NCO) are predicted using the receiver’s navigation solution. In this way,

satellites with strong signals can help track satellites with weak signals. As a result, once

a navigation solution is available, all tracking loops will remain in lock, even for those

with weak signal conditions, and short signal outages may be bridged (Lashley & Bevly

2009a; Lashley & Bevly 2009b; Spilker Jr.1996). On the other hand, since all satellites

are intimately related in VBTLs, any error in one space vehicle’s (SV’s) channel can

potentially adversely affect other channels (Petovello et al 2008a).

4

1.1.3 GPS/IMU Integration Strategies

Since GPS and INS are complementary, they can be integrated and a higher navigation

performance can be achieved (Petovello 2003; Greenspan 1996). There are three

GPS/INS (or GPS/IMU) integration strategies, namely loose, tight and ultra-tight

(Bullock et al 2006; Greenspan 1996; Gebre-Egziabher 2007). In loose integration, a GPS

receiver’s navigation solution (position and velocity output) is integrated with an INS,

thus the GPS receiver and INS can operate independently. In tight integration, a GPS

receiver’s raw data measurement (generally pseudorange and Doppler shift measurement)

is integrated with an INS. In this case, the GPS receiver does not necessarily provide

position and velocity information by itself. In the ultra-tight approach, sometimes called

deep or deeply coupled integration (Crane 2007; Soloviev et al 2004a; Gustafson &

Dowdle 2003), the GPS receiver has no individual tracking loops. The individual tracking

loops are replaced by a GPS/INS navigator or VBTLs such as a VDLL and/or VFLL

(Gebre-Egziabher 2007; Bullock et al 2006; Van Dierendonck 1996). Effectively, the

GPS and IMU are no longer independent sensors.

Further to the above, the tight integration approach is sometimes classified in the

literature into two sub-types: tight and alternative tight (Gebre-Egziabher 2007).

Sometimes the alternative tight GPS/INS is called an inertial-aided GPS receiver. The

alternative tight means the GPS receiver’s tracking loops receive Doppler aiding

information from the INS. In this dissertation, this approach is called tight with loop

aiding or tight loop aiding (TLA). Both TLA and UT integration systems have many

5

advantages, such as noise suppression, anti-jamming capability and improved navigation

performance (Lashley & Bevly 2009b; Gao 2007a; Petovello et al 2007).

As with a full IMU, a reduced IMU can also have the same three kinds of integration

strategies: loose, tight (tight and TLA), and ultra-tight. But for a GPS/reduced IMU, there

is only limited research, and most is about loose or tight integration (Xing & Gebre-

Egziabher 2009; Sun et al 2008; Niu et al 2007b). Research about TLA and UT

GPS/reduced IMU is rare (Petovello et al 2007). Compared with a full IMU, the key

difference with a reduced IMU is that since there are no horizontal gyros, the pitch and

roll cannot be calculated or observed directly from the inertial data and the navigation

performance is thus affected by local terrain variations. As a result, the integration of

GPS/reduced IMU needs to be adjusted or redesigned compared to a full IMU.

1.1.4 Navigation Filter

In GPS/INS (or GPS/IMU) integration, generally a navigation filter is employed for

blending GPS and INS information (Grewal & Andrews 2001; Farrell & Barth 1999). It

is used to estimate the errors of GPS and INS in the integration system. Once these errors

are estimated and compensated, the system’s navigation performance will be improved.

Usually a Kalman filter is used as the navigation filter because it is a time domain filter

with an iterative structure well suited for running on a computer. Since a conventional

Kalman filter is derived from linear systems, when an integration system is linear, it can

work very well. But in practical cases, as is the case for an INS, the system is rarely

perfectly linear although its error model can be approximated using linear equations. The

underlying assumption in this approach is that the attitude error of the INS is

6

appropriately small. In this light, nonlinear estimation techniques or nonlinear Kalman

filters need to be reviewed.

In nonlinear integration systems, such as tight GPS/INS and UT GPS/INS, nonlinear

Kalman filters need to be employed, such as a linearized Kalman filter (LKF), an

extended Kalman filter (EKF), and an unscented Kalman filter (UKF) (Xing & Gebre-

Egziabher 2009; Wendel et al 2005; Gelb1974). For a GPS receiver, if its measurement

output is position and velocity information, its measurement equations are linear. If the

measurements are pseudorange and Doppler shift (or in-phase I and quadraphase Q

signals), as will be discussed later in this dissertation, the measurement equations are

nonlinear. Therefore, tight and UT integration systems are nonlinear systems, and

generally an EKF is employed for data fusion. Prior to using a Kalman filter, its process

noise covariance matrix, Q , and measurement noise covariance, R , must be determined

or known. In practice, it may be difficult to know, or even suppose, these values. In this

case, an adaptive Kalman filter, whose Q and/or R can be estimated online or whose

filter gain is estimated online directly while without need to estimate the Q and/or R ,

needs to be used for data processing (Mohamed & Schwarz 1999; Gelb1974).

1.2 Overview and Limitations of Previous Work

Applying a TLA and/or UT GPS/reduced IMU in land vehicle navigation is where

current research is focused. Although there is much research about TLA and UT using

full IMUs (Sivananthan & Weitzen 2009; Lashley et al 2008a; Petovello et al 2008a;

Soloviev et al 2007), research with reduced IMUs is seldom seen. It is expected that some

research results for a GPS/ full set IMU can be applied to a GPS/reduced IMU. However,

7

for the integration of a reduced IMU, there are still many problems that remain unsolved,

such as the mechanization equation and error model choice for a reduced IMU, the PLL

noise bandwidth determination of an inertial-aided GPS receiver, and the preferable

configuration of an UT GPS/reduced IMU. This dissertation focuses on the TLA and UT

integration with a reduced IMU. As will be seen, some results for the reduced IMU can

also be applied to full IMUs, such as the PLL noise bandwidth computation for TLA

system, and the configuration design and analysis for UT system.

1.2.1 Reduced IMU

Although there are a few configurations for a reduced IMU, such as one vertical gyro

plus three or two accelerometers (Iqbal et al 2008; Niu et al 2007b; Daum et al 1994), one

vertical gyro plus one forward accelerometer (Phuyal 2004), and one vertical gyro plus

one odometer (Vlcek et al 1993), this dissertation focuses on the former since many new

model vehicles are already equipped with horizontal G-sensors and yaw rate sensors for

vehicle safety control (Gao 2007b; Rezaei & Sengupta 2007), which can be used as

accelerometers and a vertical gyro. Using the G-sensors and yaw rate sensor as a reduced

IMU can both save cost and reduce the volume of an LVNS, that is, almost no extra cost

and volume are needed for the reduced IMU.

In reduced IMUs comprised of two or three accelerometers and one vertical gyro, since

these configurations are based on the supposition that roads are relatively flat such that

horizontal gyros and (in some cases) one vertical accelerometer provide relatively little

information (Niu et al 2007b). However, in reality, these sensors do provide critical

information and their omission inevitably degrades the performance of the navigation

8

system. Specifically, if a reduced IMU has only one vertical gyro, since there are no

horizontal gyros, the pitch and roll cannot be calculated or observed directly from the

inertial data and the navigation performance is thus affected by local terrain variations.

Similarly, if fewer than three accelerometers are used, full knowledge of the vehicle’s

acceleration is unavailable. Both situations introduce errors in the navigation system.

In order to overcome the disadvantages of reduced IMUs, integrating them with other

navigation systems or constraints is a means of improving performance; for example,

GPS/reduced IMU integration (Xing & Gebre-Egziabher 2009; Nui et al 2007b; Phuyal

2004) and GPS/reduced IMU with vehicular constraints or odometer measurement (Li et

al 2009; Iqbal et al 2008; Syed et al 2007). For reduced IMUs, the focus of this

dissertation is their mechanization equations and error models, rather than the use of

vehicular constraints or the inclusion of odometer measurements. Although previous

work in this area has provided a few kinds of mechanization equations and error models,

such as dead reckoning (DR) type (Xing & Gebre-Egziabher 2009; Iqbal et al 2008;

Vlcek et al 1993) and full dimension type (Nui et al 2007b), it is hard to say that these

mechanization equations and error models can mostly or completely satisfy the

requirements of a GPS/reduced IMU for vehicle safety and control because they both

have some drawbacks which may degrade the performance for a reduced IMU-based

system. For DR type, roads are supposed to be completely flat, so pitch and roll are zero

in the mechanization equations and corresponding error model. This type imposes too

strong constraints on the model. For full dimension type, the mechanization equations

and error model of a full set IMU are used as those of reduced IMUs, so pitch and roll

9

values in the mechanization equations and error model cannot be constrained anymore.

Neither model can match the real situations that the pitch and roll of a vehicle are

determined mostly by the local terrain, which introduces errors in the navigation system.

1.2.2 GPS Receiver

In LVNS, in order to improve the tracking performance and the measurement quality of a

GPS receiver, the receiver can be aided with a full set IMU or a reduced IMU. In this

dissertation, focus is on a reduced IMU-aided GPS receiver (i.e. TLA GPS/reduced IMU)

and UT GPS/reduced IMU. For a GPS-only receiver, it is generally required to have the

abilities of weak signal tracking and anti-jamming (Yu 2007; Petovello & Lachapelle

2006a; Humphreys et al 2005; Psiaki 2001). In order to obtain such abilities, a standard

GPS receiver has to be changed to include new functions or algorithms (Im et al 2007;

Psiaki & Jung 2002). Herein, a GPS receiver obtains these abilities through inertial

Doppler aiding for PLL or UT integration, which are more efficient than the GPS-only

case in anti-jamming and weak signal tracking (Lashley & Bevly 2008a; Petovello et al

2008a; Im et al 2007). In land vehicle navigation (LVN), because the GPS receiver aided

with a reduced IMU or on-vehicle sensors has some advantages over not only a GPS-only

receiver but also loose and tight GPS/reduced IMUs, which will be discussed in this

dissertation, it can be envisioned that this kind of receiver will be popular in LVN in the

near future.

1.2.3 TLA GPS/Reduced IMU

In TLA GPS/reduced IMU, since the PLL of the GPS receiver is aided with Doppler shift

information, it offers some advantages over GPS-only receivers including a more

accurate navigation solution (especially for velocity and thus position), improved

10

tracking ability for high dynamics and enhanced anti-jamming performance (Sivananthan

& Weitzen 2009; Kim et al 2007; Petovello et al 2007; Gautier & Parkinson 2003).

In general, in any TLA GPS/IMU system (with a full set or reduced IMU), each PLL

tracking loop is controlled by a loop filter whose order and noise bandwidth impact the

performance of the tracking loop, regardless if the loop is being aided with inertial data or

not. With Doppler aiding from an inertial system, the noise bandwidth of the loop filters

of the PLLs can be narrowed more than in a GPS-only case because the inertial system

removes most of the user dynamics (Chiou et al 2008; Alban et al 2003; Jwo 2001). This

results in improved noise suppression within the receivers (Petovello et al 2007, Alban et

al 2003). However, the loop noise bandwidth cannot be made arbitrarily small and is

limited by the navigation solution error of the GPS/reduced IMU and the receiver’s

oscillator errors (Chiou 2005; Gebre-Egziabher et al 2005). To this end, for a constant

tracking loop noise bandwidth, the bandwidth is generally selected to accommodate the

largest signal dynamics resulting from the combined effect of the navigation solution and

oscillator errors. Because the navigation solution error varies with time, noise

suppression is weakened when the error of the GPS/reduced IMU is small (in comparison

to the largest expected error). This problem is exacerbated with a reduced IMU because

the roll and pitch parameters cannot be directly observed resulting in navigation solution

errors that change with the local terrain and vehicle orientation changes (Sun et al 2008).

However, there is no method or formula to calculate the noise bandwidth according to the

navigation performance of a TLA GPS/reduced IMU. Only a few works provided some

11

useful analyses for the phase error of a Doppler-aided PLL (Chiou et al 2007; Gebre-

Egziabher et al 2005), which are helpful in calculating the noise bandwidth according to

the navigation performance. As such, in a TLA GPS/reduced IMU, how to choose the

noise bandwidth is still an open problem. It is known that with Doppler aiding the noise

bandwidth of the PLLs can be reduced, but the extent to which this is possible is as yet

unknown. These are fundamental problems in the design of a TLA GPS/reduced IMU.

All previous research involving TLA systems focuses on full IMUs and research with

reduced IMUs is rare. Actually, the TLA with reduced IMU is obviously different from

the full IMU case, especially for the reduced IMU configuration with only two horizontal

accelerometers and one vertical gyro (called the 2A1G configuration). In this

configuration, the vertical acceleration cannot be measured and the user dynamics in the

vertical direction cannot be effectively removed. As a result, the noise bandwidth cannot

be narrowed as much as a TLA with a full IMU. Even for the reduced IMU configuration

with three accelerometers and one vertical gyro (called the 3A1G configuration), since

the pitch and roll parameters cannot be directly observed, the user dynamics cannot be

removed as much as with a full IMU. This is because even for the 3A1G configuration,

its attitude error is still larger than that of a full IMU with the same grade inertial sensors

resulting in larger unknown user dynamics (Sun et al 2008), thus affecting the noise

bandwidth reduction and TLA system performance.

1.2.4 UT GPS/Reduced IMU

In UT GPS/IMU integration, the tracking loops of the GPS receiver are controlled by the

navigation solution of the system thus resulting in vector tracking schemes, such as

12

VDLL for code correlation and VFLL for carrier Doppler removal (Lashley et al 2008b;

Petovello et al 2008a; Crane 2007; Spilker Jr.1996). As a result, UT GPS/IMU

integration has some potential advantages over GPS-only receivers such as a more

accurate navigation solution, improved weak signal tracking, enhanced anti-jamming

ability, and more rapid signal recovery after a satellite blockage (Kennedy & Rossi 2008;

Lashley & Bevly 2008a; Im et al 2007; Kim et al 2007). As with a full IMU, UT

GPS/reduced IMU has the same advantages, but the extent of the performance

improvement is different or low because the user dynamics cannot be removed as much

as for the full set IMU, which will be discussed in the dissertation.

In a UT GPS/reduced IMU, the tracking loops of the GPS receiver are implemented with

VBTLs. Code tracking is implemented with VDLL and carrier tracking is implemented

with either VFLL or vector phase lock loop (VPLL). Since, compared to PLL, VFLL can

decrease velocity estimation quality, and increase the complexity in estimating navigation

bits (Kiesel et al 2007), using VFLL for carrier Doppler removal is not very popular

when carrier phase lock can be obtained. Generally a VFLL or FLL is used for Doppler

removal only after acquisition but before phase lock is obtained (Ward et al 2006).

Following the VFLL or FLL the Doppler removal is usually implemented with a PLL or

VPLL. Since a VPLL has a demanding requirement for position accuracy, it is quite hard

to implement (Crane 2007). For example, for an L1 carrier signal, its wavelength is 19

cm. In order to implement a VPLL, the calculated pseudorange error for each channel

should be controlled within 19/4 cm, resulting in a similar position accuracy requirement

for a GPS/reduced IMU. Because of the difficulties in implementing a pure VPLL, an

13

approximate VPLL can be implemented with a cascaded scheme, whose carrier NCOs

are controlled by both the local filter and the navigation filter (Petovello et al 2008a;

Crane 2007; Ohlmeyer 2006).

Any GPS receiver must have both a code correlation function and a carrier Doppler

removal function (Borre et al 2007; Ward et al 2006; Misra & Enge 2001). Each of the

functions can be implemented with different tracking schemes such as a vector-based

tracking scheme (VBTS) or a scalar-based tracking scheme (SBTS). If the code

correlation function is implemented with VBTS, with different carrier Doppler removal

function implementations, the following vector-based GPS receivers are possible:

• VDLL plus VFLL (called VDF): It is straightforward to implement, but estimating

navigation bits is complicated.

• VDLL plus VPLL (called VDP): It is hard to implement.

• VDLL plus cascaded PLL (called VDCaP): It is straightforward to implement.

Because of the difficulties in implementing a VPLL, this dissertation focuses on the

integration of a GPS receiver with a VDLL plus VFLL configuration and a reduced IMU

(RIMU) (i.e. UT GPS/RIMU with VDF), and the integration of a GPS receiver with

VDLL plus cascaded PLL configuration and a reduced IMU (i.e. UT GPS/RIMU with

VDCaP).

Although there have been many investigations about UT integration of GPS with full

IMUs (Sivananthan & Weitzen 2009; Petovello et al 2008a; Crane 2007), investigations

14

with reduced IMUs are limited. Certainly, many investigations for UT with full IMUs can

provide significant references for the corresponding reduced IMU approach. However,

for UT GPS/RIMU, there is still a lot of work remaining to be done, such as researching

an innovative configuration for a vector-based GPS receiver, which can bear most

advantages of both the VFLL and cascaded PLL, research for a new strategy applied to

GPS outages to improve GPS signal recovery in GPS/RIMU, and the performance

evaluation for UT GPS/RIMU, which can be used to provide significant references for

practical LVNS development.

1.3 Research Objectives and Contributions

From the above review of GPS/IMU integration, it can be seen that previous research on

TLA and UT GPS/reduced IMU is limited. Even for a reduced IMU, its mechanization

equations and error model still need to be innovated. Therefore, the research described

herein includes reduced IMU, TLA GPS/RIMU, and UT GPS/RIMU. The focus is on

improving the navigation performance, the continuity, and the consistency of TLA and

UT GPS/RIMU, and enhancing vehicle safety and control. This latter aspect is one

which many previous studies did not consider explicitly. The difference between a TLA

and UT system in LVN is also addressed in this dissertation and comparisons for the two

systems are made, such as navigation performance comparison, system configuration

comparison, and system stability comparison.

With regard to the demands of future LVNS development and the limitations of previous

work, the main objectives of this dissertation are to develop innovative algorithms and/or

configurations for reduced IMU, TLA, and UT GPS/RIMU to improve LVNS

15

performance and to investigate different types of TLA and UT approaches to provide

helpful results for LVNS development. More specific objectives are outlined as follows:

1. Develop a set of mechanization equations and a corresponding error model for a

reduced IMU to improve the navigation performance of GPS/reduced IMU. As

stated earlier, in the reduced IMU case, because there are no horizontal gyros, the

pitch and roll cannot be calculated or observed directly from the inertial data, and

the navigation performance is thus affected by local terrain variations. However,

the GPS data inherently contains some information regarding the terrain (i.e.,

from the computation of the vehicle’s position). With this in mind, an INS error

model that incorporates the local terrain will be developed to improve the

navigation performance of the GPS/RIMU system.

2. Develop a proper PLL loop filter for a TLA GPS/reduced IMU to improve system

performance. In a TLA system, since the PLL of the GPS receiver is aided with

the Doppler shift calculated from the navigation solution of the GPS/RIMU

system (and satellite information), the noise bandwidth of the loop filters of the

PLLs can be narrowed more than in a GPS-only case. Since the loop noise

bandwidth cannot be made arbitrarily small and is limited by the navigation

solution error of the GPS/RIMU and the receiver’s oscillator errors, methods of

choosing the noise bandwidth for the PLL loop filter are investigated in order to

improve signal tracking, signal reacquisition and overall navigation solution

performance. Specifically, the bandwidth can be selected according to the

performance of the integration system and the quality of the receiver’s oscillator.

16

3. Design new algorithms/configurations for a UT GPS/reduced IMU to improve

system performance. In a UT GPS/RIMU, the tracking loops of the GPS receiver

are implemented with VBTS. For the receiver with VDCaP, code correlation is

implemented with VDLL, and Doppler removal is implemented with a cascaded

PLL. In a cascaded PLL, when the PLL is not locked, it may not provide better

Doppler measurements than an FLL discriminator. In this light, an innovative

algorithm/configuration will be developed to overcome the drawback of the

cascaded PLL and provide more reliable Doppler measurement in both the PLL

locked and PLL unlocked cases, thus improving the performance of the UT

GPS/RIMU. On the other hand, during a partial GPS outage, in which there are

less than four satellites in view, the navigation solution error of the GPS/RIMU

will increase and may become too large to maintain lock with VBTLs. In this

case, the receiver’s tracking scheme may need to be switched to a scalar or

traditional approach in order to maintain lock on the satellites still in view. As a

result, the satellites still in view can keep tracking and the performance of the

GPS/RIMU can be improved.

4. Evaluate the performance of a TLA GPS/reduced IMU. For a TLA system, with

Doppler aiding, the noise bandwidth of the Doppler-aided PLL can be narrowed

to improve tracking and navigation solution performance. These two performance

indices are investigated using many test runs with vehicle test data under different

GPS conditions, including open sky and foliage conditions. Many evaluations are

conducted to investigate the effects of IMU grades and reduced IMU

17

configurations on the performance of the TLA system, and to verify the noise

bandwidth determination for a proper PLL loop filter.

5. Evaluate the performance of a UT GPS/reduced IMU. In a UT system, with

different tracking schemes for carrier Doppler removal, different types of vector-

based receivers can be obtained, such as VDF and VDCaP. These two UT

systems are investigated first using test runs with vehicle test data. Then

evaluations are conducted to verify the new algorithms/configurations of UT

GPS/RIMU presented in item 3. Furthermore, comparisons among different

tracking schemes are made to investigate the effects of tracking schemes on the

performance of the UT systems. Finally, a UT system is compared with a TLA

system in both theory and test results to investigate differences in both their signal

tracking ability and navigation performance, which can provide useful

information for the choice of an LVNS scheme in practice.

In realizing the above objectives, several contributions are made in this dissertation. The

major contributions are summarized as follows:

1. Development of a local terrain predictor (LTP) algorithm in the GPS/RIMU to

help estimate the pitch and roll of the reduced IMU and thus to improve the

navigation performance of the GPS/RIMU in both GPS normal and outage

conditions.

2. Development of an adaptive loop filter (ALF) for Doppler-aided PLL following

the phase error analysis of the Doppler-aided PLL in a TLA GPS/RIMU. For the

18

ALF, its noise bandwidth can be adjusted according to the performance of the

integration system, the quality of the receiver’s oscillator, and the satellites

information. With the ALF, both the navigation performance and the PLL

tracking ability of the TLA GPS/RIMU can be improved.

3. Development of a composite Doppler removal loop, which is comprised of a

cascaded PLL and an FLL discriminator. This can overcome the disadvantage of

the cascaded PLL, and provide more reliable Doppler measurement in both the

PLL locked and PLL unlocked cases. Thus a vector-based GPS receiver with a

VDLL plus a cascaded PLL and an FLL discriminator (CaPF) composite loop is

obtained, which can improve the navigation performance of UT GPS/RIMU.

4. Development of a reconfigurable tracking loop for UT GPS/RIMU, which can be

applied in the partial GPS outage case. This kind of reconfigurable loop can

switch between VDLL and DLL, and switch between CaPF and PLL according to

the navigation performance of the integration system when a partial GPS outage

occurs. As a result, both the signal tracking ability and the navigation

performance of the GPS/RIMU are improved.

5. Evaluation of the performance of a TLA and a UT GPS/RIMU with vehicle test

data. Many test runs are conducted to investigate the signal tracking ability and

navigation performance of the TLA GPS/RIMU under both GPS strong and weak

signal cases. For a UT GPS/RIMU, different types of vector-based GPS receivers

are investigated, such as VDF, VDCaP, and VDLL plus a cascaded PLL and an

FLL discriminator (VDCaPF). Finally, TLA and UT integration systems are

19

compared both in theory and through test results to explore their differences in

LVN performance.

1.4 Dissertation Outline

This dissertation covers issues related to LVNS development including reduced IMU,

TLA and UT GPS/RIMU. It contains six chapters and two appendices with the remaining

chapters organized as follows.

Chapter Two gives an overview of the navigation systems related to LVN, including INS,

GPS, and GPS/INS integration systems. The basic concepts, principles, and primary

equations of these navigation systems are introduced in this chapter, which can facilitate

understanding of the contents of the subsequent chapters.

In Chapter Three, a local terrain predictor (LTP) is presented for reduced IMU error

modeling resulting in a set of innovative mechanization equations and error model for a

reduced IMU. This set of innovative mechanization equations and error model is derived

and compared in theory with other sets of mechanization equations and error model of a

reduced IMU to identify its advantages. This innovative algorithm of a reduced IMU is

fundamental to the following research on the TLA and UT GPS/RIMU. Following the

theoretical research of the reduced IMU, a vehicle test for data collection, which can be

used for the performance evaluation of different kinds of TLA and UT systems, is

introduced. Then the test results of loose GPS/RIMU with an LTP are presented to verify

the LTP algorithm. The results of other reduced IMU algorithms are also displayed in

order to compare the performance of different reduced IMU algorithms.

20

In Chapter Four, an adaptive loop filter (ALF) is presented for the Doppler-aided PLL

following the phase error analysis of the Doppler-aided PLL in the TLA GPS/RIMU. In

this chapter, the traditional PLL loop filter and phase error are introduced. Then, the

individual phase errors in the Doppler-aided PLL are analyzed, and their equations are

derived. Based on this analysis, the total phase error of the Doppler-aided PLL is

obtained and the ALF is designed. Following the ALF design, the test results of the TLA

GPS/RIMU are displayed. First, results from the data of a linear translation stage test are

shown to illustrate the performance of the TLA system in both strong and weak GPS

signal cases. The results of the TLA system from the vehicle test data are then presented,

including the results under different GPS conditions, such as open sky and foliage. The

results with different PLL noise bandwidths, such as constant and adaptive, are also given

and compared to verify the innovative ALF algorithm.

In Chapter Five, two innovative algorithms/configurations for UT GPS/RIMU are

presented. One is a composite Doppler removal loop, i.e CaPF. The other is a

reconfigurable tracking loop for UT GPS/RIMU. In order to design the innovative

algorithms/configurations, first the VDLL, VFLL, and cascaded PLL are detailed and

analyzed. Then based on the characteristics of the cascaded PLL and FLL discriminator,

the CaPF is presented. Further based on the characteristics of the VBTLs and traditional

tracking loops, the reconfigurable tracking loop is designed. Following the UT system

analysis and design, the UT and TLA systems are analyzed and compared in terms of

system configuration, stability, and navigation performance. After the theoretical analysis

21

and system design, the test results of the UT GPS/RIMU with vehicle test data are

presented. First, the results of the UT GPS/RIMU with VDF are displayed followed by

the results of the UT system with VDCaP. Then the results of the UT GPS/RIMU with

VDCaPF are given. Further with simulated partial GPS outages, the test results of the UT

system with reconfigurable tracking loops are presented. Finally the test results of the UT

and TLA systems are compared to identify the differences and similarities of the two

systems.

Chapter Six concludes the major results of the previous chapters, and makes

recommendations for future work.

In the appendices, the mechanization equations and error model of full IMU, and the

individual phase errors of Doppler-aided PLL are included.

22

Chapter Two: Overview of Land Vehicle Navigation Systems

For land vehicle navigation, many position and location methods can be used, such as

dead reckoning, wireless location, and image/vision-based location methods (e.g. Britt &

Bevly 2009; Mattern et al 2008; Soloviev 2008; Sönmez & Bingöl 2008). Dead

reckoning calculates position through continuously adding relative position changes to a

known initial position in a navigation frame, thus the subsequent positions always rely on

the previous point’s information. As a result, the position error increases with time. It is

the main drawback of the dead reckoning method. INS and systems with an odometer

plus a magnetic compass are typical examples of dead reckoning systems. In contrast to

dead reckoning, wireless location, including satellite location and terrestrial location,

directly determines absolute coordinates of an unknown position using measurements to

fixed reference points without taking into account previous positions. Thus its position

error does not increase with time, but is related to the measurement errors and the

geometry of the fixed reference points relative to the user. Unlike dead reckoning and

wireless location, the image/vision-based navigation approach is closely related to

photogrammetry (i.e. surveying using photographic imagery) in geometric fundamentals

(Hofmann-Wellenhof et al 2003). Since different positioning methods have different

error characteristics, navigation systems such as INS and GPS, developed from different

positioning methods, have correspondingly different position error characteristics. In

order to overcome the drawback of each individual navigation system, integrated systems

such as GPS/INS are usually employed, thus achieving more accurate and reliable

navigation performance. Based on the focus of this dissertation, GPS, INS, and their

integration are introduced in this chapter.

23

2.1 Reference Frames

Prior to introducing navigation systems, it is necessary to define a series of reference

frames that are commonly used in land vehicle navigation systems. The following

coordinate frames are used in this text:

• Inertial frame ( i -frame), denoted as ( , , , )io x y z , is a fixed coordinate frame with

the centre of the Earth as its origin io . Its z axis, iz , is parallel to the spin axis of

the Earth, ix points to the mean vernal equinox, and iy is determined by ix and iz

in a right-handed system.

• Earth-fixed frame ( e -frame), denoted as ( , , , )eo x y z , coincides with the i-frame at

the origin but rotates with the Earth rate. Its z axis, ez , is parallel to iz , ex points

to the mean meridian of Greenwich, and ey is determined by ex and ez in a right-

handed system.

• Local level frame ( ℓ -frame), denoted as ( , , , )o x y zℓ

, is an east, north, up

rectangular coordinate system, which is called ENU. Its origin, oℓ, is at the

location of the navigation system and on the Earth’s surface, xℓ points east, y

ℓ

points north, and zℓ points vertically upwards.

• Body frame ( b -frame), denoted as ( , , , )bo x y z , is rigidly attached to the vehicle of

interest, usually at a fixed point such as the centre of gravity. The bx , by and bz

axes point in the right, forward and up directions, respectively.

• Alternative level frame ( 1ℓ -frame), denoted as 1( , , , )o x y zℓ

, is another local level

frame which has the same origin and vertical axis with those of ℓ -frame, but the

24

directions of other two axes ( 1xℓ

and 1yℓ

) are determined by the directions of the

corresponding two axes bx and by . This frame is used as an intermediate frame to

link the local terrain (pitch and roll) and the attitude error of a reduced IMU.

Corresponding to the frames, some quantities are often used in the following equation

derivations, and defined as: ar denotes a vector in frame a, b

aR denotes a rotation matrix

from frame a to b, and a

pqω denotes a vector of rotation rates of frame q, relative to frame

p, expressed in frame a.

2.2 Inertial Navigation System

The operation of inertial navigation is based on the laws of classical mechanics as

formulated by Sir Isaac Newton (Rana & Joay 2006). Newton’s first law states that a

body in motion tends to maintain its motion unless disturbed by an external force acting

on the body. His second law states that this force produces a proportional acceleration of

the body. If the acceleration can be measured and converted to a navigation frame with

appropriate transformations, then in the navigation frame, a single integration yields

velocity and a second integration provides change in position. Acceleration can be

determined using a device known as an accelerometer, and an INS usually contains three

such devices mounted orthogonally. In order to obtain appropriate orientation

transformations for acceleration, gyroscopic sensors (or simply gyros) are needed.

Generally, three orthogonal gyros are needed to obtain full knowledge of the orientation

of accelerometers. A full six-degree of freedom IMU (“full IMU”) therefore consists of

three orthogonal accelerometers and three gyros. On the other hand, sometimes less than

25

three gyros and/or accelerometers can be used for navigation, forming a reduced IMU.

Both of the systems are introduced in the following sections.

2.2.1 Full IMU System

A full IMU consists of three orthogonal accelerometers and three orthogonal gyros.

These accelerometers and gyros can be mounted together on either a platform or a vehicle

directly, resulting in two different kinds of navigation systems: gimballed and strapdown.

This dissertation focuses on the latter.

In a strapdown system, the sensor assembly is directly mounted onto the vehicle and

follows all motions performed by the vehicle reference. The accelerometers measure the

specific force bf (i.e., force per unit mass) along the axes of the body frame, whereas the

gyros sense the angular rate b

ibω of the body frame relative to the inertial frame. These

measurements are used to calculate the vehicle’s navigation information with a set of

navigation equations. Navigation equations of a system can be derived in different frames

– herein called the navigation frame – such as the inertial frame, Earth-fixed frame, and

local level frame. Since this dissertation focuses on land vehicle navigation, in order to

facilitate analysis, the local level frame is chosen as the navigation frame.

2.2.1.1 Navigation Equations

In order to derive the navigation equations in the local level frame, the navigation

equations in the inertial frame need to be introduced first. The fundamental relationship is

the specific force equation in the i-frame:

i i i= +r f Gɺɺ (2.1)

26

where, irɺɺ is the acceleration of the vehicle, r is the position vector of the point P with

respect to io shown in Figure 2.1, if is the specific force, and iG is the gravitational

acceleration. The first integral of Equation (2.1) gives the velocity vector iv , and

i i i

i i

= +

=

v f G

r v

ɺ

ɺ (2.2)

Equation (2.2) contains two linear differential equations with constant coefficients. Using

these, velocity and position can be computed.

Figure 2.1 Position Vector in Inertial Frame

Based on the above navigation equations in the i -frame, the corresponding equations in

the ℓ -frame can be derived. However, additional steps are required. Effectively, in order

to obtain the navigation equations in the ℓ -frame, the specific-force equation in the e -

frame is developed first, and then transformed to the ℓ -frame. This procedure is applied

as follows.

From the relationship between the position vector in the e -frame and i -frame:

oi

zi

yi

xi

P

27

i i e

e=r R r (2.3)

where i

eR is the transformation matrix or rotation matrix, which rotates the e -frame into

i -frame. And the i

eR is determined as the solution to the differential equation

i i e

e e ie=R R Ωɺ (2.4)

where e

ieΩ is the skew-symmetric matrix

e e

ie ie = × Ω ω (2.5)

This matrix is formed from the elements of the vector e

ieω which represents the turn rate

of the e -frame with respect to the i -frame. If ( )T

a a a a

pq pqx pqy pqzω ω ω=ω ,

0

0

0

a a

pqz pqy

a a a

pq pqz pqx

a a

pqy pqx

ω ω

ω ω

ω ω

−

= − −

Ω . Through a series of derivations, the acceleration vector

erɺɺ is obtained (Hofmann-Wellenhof et al 2003)

2e e e e e

ie= + −r g f Ω rɺɺ ɺ (2.6)

where ef is the specific force vector in the e -frame, eg is the gravity vector and equal to

the gravitational vector eG minus the centrifugal acceleration e e e

ie ieΩ Ω r , i.e.

e e e e e

ie ie= −g G Ω Ω r (2.7)

Subsequently, Equation (2.6) is transformed to the ℓ -frame using the relationship

between vℓ and erɺ

e e=r R vℓ

ℓɺ (2.8)

28

Finally, the navigation equations in the ℓ -frame are obtained as (Titterton & Weston

2004; Hofmann-Wellenhof et al 2003)

1

2( )i ie

−

= + − +

=

v g f Ω Ω v

r D v

ℓ ℓ ℓ ℓ ℓ ℓ

ℓ

ℓ ℓ

ɺ

ɺ

(2.9)

where 1−D is a matrix which transfers velocity in the ℓ -frame into a position variation

rate in the ℓ -frame (see Appendix A), and the position is expressed with curvilinear

geodetic coordinates (latitude, longitude, and height). The specific force, f ℓ , can be

obtained from the accelerometer measurement bf , namely

b

b=f R fℓ ℓ (2.10)

where b

Rℓ is the rotation matrix describing the attitude of the vehicle. It is computed from

the gyro data using numerical integration of (Hofmann-Wellenhof et al 2003)

b

b b b=R R Ω

ℓ ℓ

ℓɺ (2.11)

The detailed navigation equations are shown in Appendix A.

2.2.1.2 Error Model

In inertial systems, navigation error is induced by a variety of factors such as the inertial

sensors, numeric computations, and initial values. Among these sources, the inertial

sensors introduce the main error. In order to understand the navigation performance of

the INS, the dynamic behaviour of the navigation system error is further analyzed below.

In the ℓ -frame, perturbing the second equation of Equation (2.9) yields the linearized

position error dynamics (e.g. El-Sheimy 2007; Farrell & Barth 1999)

1 1

rδ δ δ− −= − +r D D r D vℓ ℓ ℓɺ (2.12)

29

where r

D is a coefficient matrix, and 1

r

−−D D is defined in Appendix A. In a similar way,

perturbing the first equation of Equation (2.9) yields the linearized velocity error equation

(El-Sheimy 2007; Farrell & Barth 1999)

b

bδ δ δ δ δ= + − + +v A r B v F ε R f g

ℓ ℓ ℓ ℓ ℓ ℓ ℓɺ (2.13)

where A , B , and Fℓ are matrices defined in Appendix A, and Fℓ is the skew-symmetric

matrix form of the specific force vector f ℓ . εℓ is attitude error, bδ f is accelerometer

measurement error, and δ gℓ is the gravity computational error. From Equation (2.13), it

can be seen that the velocity error variation is not only related to the position and velocity

error, but also related to attitude error, specific force, and accelerometer measurement

error.

The attitude error is defined as the misalignment caused by measurement, computational,

and initialization errors. For small angles of misalignment, the computed transformation

matrix can be written as (Titterton & Weston 2004)

( )b b

= +R I E Rℓ ℓ ℓɶ (2.14)

where I is a 3 3× unit matrix, Eℓ is the skew-symmetric matrix of the attitude error εℓ ,

and b

Rℓ is the true transformation matrix. Starting from Equation (2.14), through a series

of differentiation and approximation operations, the attitude error equation is obtained as

(Titterton & Weston 2004)

l l l l l l b

il b ibδ δ δ= + − +ε P r Q v Ω ε R ωɺ (2.15)

30

where matrices P and Q are defined in Appendix A, and b

ibδω is the gyro measurement

error. From Equation (2.15), it can be seen that the attitude error variation is induced not

only by the gyro measurement error but also by the position, velocity, and attitude errors.

In fact, for an INS, the gyro measurement error is the main factor causing attitude errors,

especially for low accuracy IMUs such as micro-electro-mechanical systems (MEMSs).

The attitude error and accelerometer measurement error are the main sources inducing

velocity errors. Therefore, it is important to model the sensor errors and include them in

INS error model to improve the navigation performance. Generally, inertial sensor errors

include some or all of the following components (Titterton & Weston 2004): a) fixed or

repeatable terms; b) temperature-induced variations; c) switch-on to switch-on variations;

d) in-run variations (i.e. variations with time or vehicle’s motions). A detailed error

model may contain many terms and be too complicated to use in practical systems. As a

result, in general a simplified error model of inertial sensors is often used. For instance,

the error model for both gyros and accelerometers is expressed as the sum of four terms:

a first-order Gaussian Markov (GM) process (in-run variations), white noise (in-run

variations), random constant (switch-on to switch on variations), and a scale-factor error

(in-run variations) (Godha 2006; Grewal et al 2001). To reduce the Kalman filter

computation of a GPS/INS system, the error model of inertial sensors can be further

simplified as a first-order GM process plus a white noise component (Godha 2006). In

this case, the gyro error model (for one axis) is expressed as

31

b

ib

d

d w

d d w

ωδω

α

= +

= − +ɺ (2.16)

where b

ibδω is the gyro measurement error, d is gyro drift, wω is gyro measurement noise,

α is the inverse of the correlation time of the process, and d

w is the driving noise.

Similarly, the accelerometer error model is written as

b

f

b

f b w

b b w

δ

β

= +

= − +ɺ (2.17)

where bfδ is the accelerometer measurement error, b is the accelerometer bias, b

w is

white noise, β is the inverse of correlation time, and b

w is the driving noise. Further

details on the INS error model equations are shown in Appendix A.

2.2.2 Reduced IMU System

A reduced IMU consists of less than three gyros and/or less than three accelerometers,

resulting in incomplete acceleration and/or rotation rate measurements. As a result, the

navigation performance of a reduced IMU is degraded compared to a full IMU with the

same grade inertial sensors (Sun et al 2008; Nui et al 2007b). For land vehicle navigation,

usually one vertical gyro and two or three accelerometers are used in a reduced IMU as

shown in Figure 2.2 (Sun et al 2008; Nui et al 2007b), producing two reduced IMU

configurations: 3A1G and 2A1G. These two configurations are based on the assumption

that roads are relatively flat such that horizontal gyros and (in some cases) one vertical

accelerometer provide relatively little information (Niu et al 2007b). But in practice,

roads are not flat. Slopes and tilts always exist, inevitably introducing navigation errors

32

with a reduced system. The reason that slopes and tilts introduce navigation errors in a

reduced IMU is explained in the following.

Figure 2.2 Reduced IMU Configuration

In inertial navigation, originally, a gimballed platform was used to carry the sensor

assembly in order to isolate the assembly from the angular motions of the vehicle

(Hofmann-Wellenhof et al 2003). The gyros of the assembly track the rotations of the

platform and drive servomotors to align the platform to the local level frame (if the local

level frame is the navigation frame). If the platform and the local level frames are

completely coincident, the acceleration in each axis of the ℓ -frame can be exactly

measured by the corresponding accelerometers of the assembly (without considering the

sensor errors). Otherwise the acceleration in the ℓ -frame cannot be completely measured

by the corresponding accelerometers. The accelerometer measurement has error induced

by the cross product of the specific force and the attitude error vectors; the bigger the

attitude error, the greater the accelerometer measurement error, and the greater the

navigation error. Similar conclusions can be obtained from strapdown systems. In a

Accelerometer

Up

Forward

Right

Gyro

33

strapdown system, the platform is implemented with a rotation matrix (i.e., b

Rℓ if ℓ -

frame is navigation frame). For a reduced IMU, since there are no horizontal gyros, its

platform (or b

Rℓ ) cannot be adjusted to be completely coincident with the ℓ -frame.

Actually, the platform mostly follows the local terrain around its two horizontal axes. As

a result, the attitude error of the reduced IMU depends on local terrain variations, and so

does the navigation error. Since reduced IMU displays different characteristics from the

full IMU, its mechanization equations and error model need to be discussed, as shown in

Chapter Three.

2.3 GPS Receiver

A GPS receiver is a device that receives GPS signals for determining the user’s position

and velocity. Generally, a GPS receiver consists of five parts or functions: GPS signal

receiving, signal conditioning, signal acquisition, signal tracking, and navigation

processing (Borre et al 2007; Misra & Enge 2001). Although each part is important,

based on the focus of this dissertation, the following mainly introduces the last two parts.

2.3.1 GPS Receiver Configuration

A generic GPS receiver functional configuration is shown in Figure 2.3 (Ward et al 2006;

Raquet 2004). In this figure, RF denotes radio frequency, IF denotes intermediate

frequency, and LO denotes local oscillator. Figure 2.3 contains a GPS antenna, front end

part, an oscillator, signal processing part, and navigation processing part. Among these

parts, the antenna is for GPS signal receiving. After signal receiving, the front end part

implements signal conditioning, including signal amplification, frequency down

conversion, and signal sampling. Then the signal processing part implements signal

acquisition and tracking functions. Herein, the signal acquisition carries out a global

34

search for approximate values of the code phase and Doppler shift. After the acquisition

algorithm has significantly reduced the code and Doppler errors, signal tracking

commences. The main purpose of tracking is to refine the code phase and Doppler shift

values, continue to track the signals, and demodulate the navigation data. Signal tracking

can provide pseudorange, Doppler shift, and carrier phase measurements. Once these

measurements from at least four satellites are available, the navigation processing part

starts the calculation of the user’s position, velocity and time (PVT). During the above

processes, the oscillator and frequency synthesizer generate signals to control the

frequency down conversion and signal processing. Furthermore, the user can control both

the signal processing and navigation processing through interrupts. Generally, a GPS

receiver contains several channels, such as 12 channels. Each channel consists of the

circuitry or software (for software receiver) necessary to track the signal from one single

GPS satellite.

Figure 2.3 GPS Receiver Functional Configuration

35

2.3.2 GPS Receiver Scalar-Based Tracking Loops

GPS signal tracking can be implemented with either a scalar-based tracking scheme

(SBTS) or a vector-based tracking scheme (VBTS). With SBTS, each channel of the

receiver has its own tracking loops that operate independently of other channels (i.e., the

loops are closed locally). Accordingly, the locally-closed tracking loops are called scalar-

based tracking loops (SBTLs). Generally, with SBTS, each channel has two of this kind

of locally-closed SBTLs: one is a delay lock loop (DLL), the other is a carrier tracking

loop (CTL) or carrier Doppler removal loop consisting of one or both of a phase lock

loop (PLL) and a frequency lock loop (FLL). In order to implement both code and carrier

wipeoff functions, a GPS receiver must have both a DLL and a CTL (either a PLL or a

FLL, or both), as shown in Figure 2.4 (Ward et al 2006; Raquet 2004). In the figure, “E”

stands for “early”, “P” stands for “prompt”, and “L” stands for “late”. I is in-phase signal

and Q is quadraphase signal. Specifically, Figure 2.4 illustrates that the input signal is

first multiplied with a carrier replica to wipe off the carrier wave from the signal

(“Doppler Removal”); then the signal is multiplied with a code replica to remove the

code from the signal (“Correlation”), and provides the navigation message obtained from

I3j(P) (for more information, please see Borre et al (2007) ). In this dissertation, the

pseudo-random noise (PRN) code is C/A code, and the carrier wave is 1L GPS signal

with the frequency 1 1575.42L

f = MHz.

36

Figure 2.4 Scalar-Based Tracking Loops of GPS Receiver (One Channel)

In a CTL, when there is excess Doppler error, generally an FLL operated with a wide

bandwidth is used because the FLL is more robust (Ward et al 2006). With the Doppler

error decreasing, the FLL will gradually reduce its bandwidth and finally transition into a

wideband PLL. Subsequently, the PLL will gradually narrow its bandwidth to the steady

state model of operation. PLLs are considered as the desired steady state tracking model

because they can not only produce the most accurate Doppler measurements, but also

provide the most error-free data demodulation compared to the FLLs (Ward et al 2006).

However, the FLLs fulfill the carrier wipeoff process by replicating the approximate

frequency. They have wider frequency pull-in ranges, and are easier to acquire lock

compared to the PLLs. So they are preferable in cases such as large Doppler error and

high dynamics, where it is difficult to acquire and maintain signal lock using PLLs.

37

Both the DLL and the CTL are sophisticated systems, but either system can be modeled

as a control system to facilitate system analysis (Misra & Enge 2001). In particular, for

both a DLL and a Costas PLL, either system is modeled as a linear phase lock loop like

the one shown in Figure 2.5 (Alban et al 2003; Misra & Enge 2001). In Figure 2.5,

( )R s is the input phase (either carrier or code), ˆ ( )R s is the output phase,

ˆ( ) ( ) ( )E s R s R s= − is the actuating error, )(sN is noise or disturbance, ( ) 1 /pG s s= is

the transfer function of the NCO, ( )F s is the Doppler shift and ( )c

G s is the controller

or loop filter. For a Doppler-aided DLL, generally a first-order loop filter is used, i.e.

( )c p

G s k= , where pk is a constant (Ward et al 2006). For a Doppler-aided PLL, usually a

second-order loop filter is used (Ward et al 2006), i.e. ( )2( ) 2c n n

G s s sξω ω= + , where

707.0=ξ is the damping ratio of the system and nω is the un-damped natural frequency.

Without Doppler aiding, the order of the loop filters should be higher than that with

Doppler aiding (Ward et al 2006; Babu & Wang 2005). For an FLL, just like the PLL, it

can be modeled as a control system as well, and has a similar transfer function as the PLL

if the input signal of the FLL is frequency.

Figure 2.5 Block Diagram of Linearized Phase Lock Loop (One Channel)

38

In SBTLs, the measurement error or tracking error of a loop can be induced by numerous

sources such as the loop’s nonlinearity and the receiver’s dynamics (Ward et al 2006).

These sources induce measurement errors in the following way: first each of the sources

produces an input error or noise; then this input error propagates in the loop, generating a

measurement error. Consequently, the magnitude of this measurement error is not only

related to the magnitude of the input error, but also related to the loop’s noise bandwidth.

In a PLL, there are numerous sources of phase measurement errors. However, it is

sufficient to consider only the dominant error sources such as thermal noise, oscillator

noise, and receiver dynamics. The formulae calculating these source-induced phase errors

are given in Appendix B.

2.3.3 GPS Receiver Vector-Based Tracking Loops

When VBTS is applied in GPS signal tracking, the tracking loops are termed vector-

based tracking loops (VBTLs). In contrast to SBTLs, VBTLs of a GPS receiver are

closed via the navigation solution coming from either the GPS-only or the GPS/INS

integration system. Thus the tracking loops of each channel are not independent, i.e. the

loops of different channels are coupled through the navigation solution or navigation loop.

Like SBTLs, there are also two types of VBTLs: one is vector delay lock loop (VDLL),

the other is vector carrier tracking loop (VCTL) consisting of one or both of a vector

phase lock loop (VPLL) and a vector frequency lock loop (VFLL).

A vector-based GPS receiver can be composed of a VDLL and either a VCTL or a scalar

carrier tracking loop (SCTL) (or simply CTL). An example of a vector-based GPS

receiver consisting of a VDLL and a VFLL is shown in Figure 2.6. From Figure 2.6, it

39

can be seen that in each channel, both the code NCO and the carrier NCO are controlled

by the output of the GPS/INS integration system. The pseudorange error and carrier

Doppler error directly come from the corresponding discriminators. There are no locally-

closed tracking loops. Correspondingly the noise in a tracking loop does not propagate

through a locally-closed loop.

Figure 2.6 Vector-Based Tracking Loops of GPS Receiver (One Channel)

Like SBTLs, VBTLs also can be modeled as a linear control loop for one channel, but

this loop is broken or open. Figure 2.7 takes a VDLL as an example to illustrate the block

diagram of a VBTL. From Figure 2.7, it can be seen that the code NCO is controlled with

Doppler aiding obtained from the navigator of the receiver, and reset frequently with the

code phase that is also obtained from the navigator. The code phase measurement

ˆm

τ equals the output of the NCO plus the filtered output of the code discriminator, and m

τ

is the input code phase. Figure 2.7 is a discrete system, wherePIT

T is the predetection

40

integration time (PIT). This figure is drawn with reference to the block diagram of the

discrete linearized model of a DLL as shown in Misra & Enge (2001).

Figure 2.7 Block Diagram of VDLL’s Lock Loop (One Channel)

2.3.4 GPS Receiver Measurements

Generally, a GPS receiver can provide two types of measurements: one from code

tracking called a pseudorange, the other from carrier phase tracking called the carrier

phase. At the same time, carrier phase tracking gives an estimate for pseudorange rate,

called Doppler frequency or Doppler shift. In a GPS receiver, the basic measurement is

the transit time of the signal from a satellite to the receiver. The corresponding

pseudorange is defined as the transit time multiplied by the speed of light. Since there are

various factors degrading the measurement accuracy of the transit time, the pseudorange

measurement inevitably contains errors. Accounting for various measurement errors, the

measured pseudorange can be written as (Lachapelle 2006; Misra & Enge 2001)

( )u sr O c t t I Tρ ρ ρ ρρ δ δ ε= + + − + + + (2.18)

41

where the unit of ρ is metres, Oρ is orbital error, u

tδ and s

tδ are receiver and satellite

clock offsets respectively (herein advance is positive; delay is negative), c is the speed of

light, Iρ and Tρ are ionospheric and tropospheric delays respectively, ρε is noise, r is

geometric range, written as

2 2 2( ) ( ) ( )u u ur X x Y y Z z= − + − + − (2.19)

where ( , , )u u ux y z is the position of the user, ( , , )X Y Z is the position of a satellite, which

is calculated from the ephemeris.

The carrier phase measurement is much more precise than code phase, but is an

ambiguous measurement of the signal transit time. Like code phase, carrier phase

measurements also contain errors. Accounting for various measurement errors, the

carrier phase measurement can be expressed as (Lachapelle 2006; Misra & Enge 2001)

( )u sr O c t t M I Tρ ρ ρ φφ δ δ λ ε= + + − + − + + (2.20)

where the unit of φ is metres, λ is wavelength, M is integer cycle ambiguity, φε is

noise, and the remaining parameters have the same meaning as in Equation (2.18). In the

CTL of a GPS receiver, the Doppler shift is measured continuously. It is produced by

both the relative motion of the satellite with respect to the user and the receiver clock

drift. Its measurement can be written as (Kaplan et al 2006; Misra & Enge 2001)

( )ud T T u d

f f f tc

ε− ⋅

= − − +V v κ

ɺ (2.21)

where the unit of d

f is Hz, V is the satellite velocity vector, u

v is the user velocity

vector, κ is the user-to-satellite line-of-sight unit vector, T

f is the transmitted signal

42

frequency (for 1L signal, 1 1575.42L

f = MHz), u

tɺ is receiver clock drift rate (herein it is

positive if the user clock is running fast), and d

ε is noise.

2.3.5 Estimation of Position, Velocity, and Time

Once GPS measurements are obtained, the user’s position, velocity, and time can be

estimated. In the following, only pseudorange and Doppler shift measurements are used

to estimate the PVT. First, from the pseudorange measurement, the user’s position and

receiver clock offset can be obtained. From Equation (2.18), most errors such as satellite

clock offset, ionospheric delay, and tropospheric delay can be modeled and corrected first.

After the correction, substituting Equation (2.19) into Equation (2.18) yields the

pseudorange measurement of ith satellite

2 2 2( ) ( ) ( )i i u i u i u u iX x Y y Z z c t ρρ δ ε= − + − + − + + ɶ (2.22)

where iρεɶ is the noise after the pseudorange error correction. In order to determine the

user’s three-dimensional position ( , , )u u u

x y z and the clock offsetu

tδ , at least four

satellites’ pseudorange measurements are needed. With four measurements, four

corresponding equations can be obtained with Equation (2.22). Then the position and

clock offset can be solved.

A simple approach to solving the four equations is to linearize them about an

approximate user position, and solve iteratively. Let ˆ ˆ ˆ ˆ( )T

u u u ux y z=x and

utδ be the

estimates of the position and clock offset. Approximating the pseudorange equation with

a first-order Taylor series yields

43

ˆ ˆ ˆˆ

ˆ ˆ î u i u i u

i i u u u b i

i i i

X x Y y Z zx y z c t

r r rρρ ρ δ δ δ ε

− − −− = − − − + ∆ + ɶ (2.23)

where 1,2, ,i N= ⋯ , where N is the number of satellites, û u u

x x xδ = − , û u u

y y yδ = − ,

û u u

z z zδ = − , ˆb u u

t t tδ δ∆ = − , î

ρ is expressed as

ˆˆ î i u

r c tρ δ= + (2.24)

and i

r is

2 2 2ˆ ˆ ˆ ˆ( ) ( ) ( )i i u i u i ur X x Y y Z z= − + − + − . (2.25)

Once at least four linearized equations as Equation (2.23) are obtained, the four

unknownsu

xδ , u

yδ , u

zδ , and b

t∆ can be solved using a least-squares algorithm (Misra

& Enge 2001; Axelrad & Brown 1996). Finally, the user’s position and the receiver clock

offset can be obtained.

Similar to the position estimation with pseudorange, velocity and clock drift can be

estimated with Doppler shift. Starting from Equation (2.21) , the pseudorange rate is

defined as

d Tc f fρ = − ⋅ɺ (2.26)

Then the measurement of pseudorange rate for ith satellite is written as

( )i i u i u ic t ρρ ε= − ⋅ + ⋅ −V v κɺ

ɺɺ (2.27)

44

where i di Tc fρε ε= ⋅ɺ

, ( )T

i ix iy izV V V=V , ( )

T

u ux uy uzv v v=v , and

ˆ ˆ ˆ

ˆ ˆ ˆ

T

i u i u i ui

i i i

X x Y y Z z

r r r

− − −=

κ , the parameters in iκ are defined in Equation (2.23).

Rearranging Equation (2.27) yields

i i i u i u ic t ρρ ε− ⋅ = − ⋅ + ⋅ −V κ v κɺ

ɺɺ (2.28)

When at least four pseudorange rate equations as Equation (2.28) are obtained, the four

unknownsux

v , uyv , uz

v , and u

tδ ɺ can be solved with least-squares algorithm. If the user’s

dynamics is modeled as a linear equation, PVT can be solved using a Kalman filter with

the measurements of pseudorange and pseudorange rate (Axelrad & Brown 1996).

2.4 GPS/IMU Integration System

As mentioned in Chapter One, there are three kinds of GPS/INS integration systems,

namely loose, tight and ultra-tight integration. And for tight integration, the system is

further divided into two sub-types: tight and TLA system (see Chapter One). These

systems are introduced in the following.

2.4.1 Loose Integration System

In loose GPS/INS integration, the GPS receiver outputs position and velocity information

which is integrated with the INS as shown in Figure 2.8. As shown, A is attitude, INS

δ X

denotes the INS error state.

45

Figure 2.8 Loose GPS/INS Integration Block Diagram

From Figure 2.8, it can be seen that the GPS receiver has an extended Kalman filter

responsible for navigation processing. The integration filter implemented with an

extended Kalman filter is used for the information fusion of the integration system. The

output of the integration filter is used to correct the INS with a closed-loop scheme

(through ˆINS

δ X ). Since the measurements of the integration filter are position and

velocity error, the measurement equation is linear and written as

1 1 1k k k+ + += +Z HX η (2.29)

46

where 1 1 1( )T

k k kδ δ+ + +=Z P V , 1k+η is measurement noise, 1k+X is the state vector, and

1 ( 1)k INS

kδ+ = +X X . 3 3 3 3 3 ( 6)

3 3 3 3 3 ( 6)

n

n

× × × −

× × × −

=

I 0 0H

0 I 0, where 3 3×I is a 33× unit matrix, 33×0 is

a 33× zero matrix, n is the dimension of the state. If the state equation is expressed as

1k k k k k+ = +X Φ X G W (2.30)

an extended Kalman filter can be obtained (Gelb 1974). In Equation (2.30), kW is the

state process noise. Since Equation (2.30) is a linearized equation (Titterton & Weston

2004) and the INS error correction in the integration system is implemented with a

feedback loop or closed loop, the implemented integration filter is an extended Kalman

filter (Grewal & Andrews 2001; Farrell & Barth 1999).

2.4.2 Tight and Ultra-Tight Integration Systems

In tight GPS/INS integration, the GPS receiver outputs raw data measurements, i.e.

pseudorange, carrier phase, and carrier Doppler. Generally pseudorange and carrier

Doppler measurement are used to integrate with the INS as shown in Figure 2.9 (if carrier

phase is to be used, its cycle ambiguities need to be solved). In Figure 2.9, GPSδ X

denotes the error state of the GPS receiver. The integration filter is an extended Kalman

filter. The measurement equation is nonlinear and written as

1 1 1( )k k k+ + += +Z h X η (2.31)

where 1 1 1( ( 1) ( 1) ( 1) ( 1))T

k N Nk k k kδρ δρ δρ δρ+ = + + + +Z ɺ ɺ⋯ , where N is the

tracked satellite number, k denotes discrete time, ˆi i iδρ ρ ρ= − and i i i iδρ ρ= − ⋅V κɺ ɺ ,

1,2, ,i N= ⋯ , are expressed as Equations (2.23) and (2.28) respectively. If the state

equation is expressed as Equation (2.30), and the state

47

( )1 ( 1) ( 1)T

T T

k INS GPSk kδ δ+ = + +X X X , an extended Kalman filter can be obtained (Gelb

1974). In tight integration system, since both the dynamics and measurement equations of

the integration filter are nonlinear, for Kalmen filter implementation, one would have to

use extended Kalman filtering (Grewal et al 2001).

Figure 2.9 Tight GPS/INS Integration Block Diagram

The block diagram of TLA GPS/INS integration is shown in Figure 2.10. As discussed in

Chapter One, in the TLA system, the PLLs of the GPS receiver accept Doppler shift

aiding calculated from the navigation solution of the integration system and satellite

information. Comparing Figure 2.10 and Figure 2.9, it can be seen that the only

difference between the TLA and tight systems is that in the TLA system, the GPS

receiver’s PLLs are aided with the Doppler shift.

48

Figure 2.10 TLA GPS/INS Integration Block Diagram

The block diagram of UT GPS/INS integration is shown in Figure 2.11. From Figure 2.11,

it can be seen that the tracking loops of the receiver are vector-based whereas those in

TLA and tight system are scalar-based. The code phase and Doppler shift calculated from

the INS and satellite ephemeris are used to control both the code NCO and carrier NCO

to implement vector-based tracking. In the UT system, the pseudorange and carrier

Doppler are used as the measurements in the integration system (Lashley & Bevly 2008a;

Pany et al 2005). Sometimes the in-phase I and quadraphase Q signals of the receiver,

which are used as the input of the discriminators in a scalar-based receiver, are used as

measurements for a UT system directly (Crane 2007; Ohlmeyer 2006). The I and Q

signals are generated in the front-end part of any GPS receiver. In this part, the GPS

49

signal is down-converted and separated into two constituents: in-phase I and quadraphase

Q. These two components are orthogonal to each other and used in the signal acquisition

and tracking part of the receiver (Lachapelle 2006; Ward et al 2006).

Figure 2.11 UT GPS/INS Integration Block Diagram

50

Chapter Three: Mechanization of Reduced IMU

As discussed in Chapter Two, generally there are two reduced IMU (RIMU)

configurations used in land vehicle navigation, namely 3A1G (i.e. three accelerometers

plus one vertical gyro) and 2A1G (i.e. two horizontal accelerometers plus one vertical

gyro). Since there are no horizontal gyros in the reduced IMUs, their rotation

measurements are incomplete (for the 2A1G case, so is the acceleration measurement),

resulting in degraded navigation performance (Sun et al 2008; Niu et al 2007b). Owing to

the fact that reduced IMUs display different characteristics from full IMUs, their

mechanization equations and error models are reviewed in this chapter.

This chapter begins with the development of a local terrain predictor (LTP) method for an

RIMU to improve the navigation performance of a GPS/RIMU, resulting in a set of

innovative mechanization equations and error model. This set of equations and error

model and other two sets of mechanization equations and error model for RIMU (i.e.

dead reckoning and full dimension type) are compared to identify their advantages and

disadvantages. Following the theoretical research, a field vehicle test, conducted to

collect IMU and GPS data for post-mission processing, is described. An evaluation of the

LTP method is then performed with the field data, and finally a comparison of the three

sets of mechanization equations and error model is made with tests using the real data.

3.1 Mechanization of RIMU with a Local Terrain Predictor

In inertial navigation, a full six degree of freedom IMU consists of three orthogonal

accelerometers and three orthogonal gyros. As a result, the acceleration and rotation of

the vehicle mounted with the IMU are completely measured. By contrast, in a reduced

51

IMU where there is only one vertical gyro, the pitch and roll cannot be measured.

Similarly, if fewer than three accelerometers are used, such as in the 2A1G configuration,

full knowledge of the vehicle’s acceleration is unavailable. Both situations introduce

errors in the navigation system.

In order to overcome the disadvantages of RIMUs, integrating them with other navigation

systems or constraints is a means of improving performance. For example, GPS/RIMU

(Niu et al 2007b; Phuyal 2004) and GPS/RIMU with vehicular constraints (Li et al 2009;

Syed et al 2007) have previously been studied. Herein, a local terrain predictor is

presented to help estimate the pitch and roll of the RIMU and thus to improve navigation

performance. The LTP method can be envisaged as an error modeling method for RIMUs

that compensates for terrain-induced pitch and roll variations. It does not increase the

measurement variables of GPS/RIMU, but changes the navigation equations and error

model of the RIMU. Since this dissertation only focuses on two RIMU configurations, i.e.

3A1G and 2A1G, with the LTP, the navigation equations and error models of the two

configurations are investigated below.

3.1.1 Derivation of Navigation Equations of RIMU

The derivation of navigation equations is based on the 3A1G configuration, and can be

developed from the navigation equations of a full IMU as shown in Equation (A.1) in

Appendix A. Furthermore, only the attitude equation needs to be derived, as below. For

details on notation, please refer to Appendix A. The following development closely

follows that in Sun et al (2008) and as such many of the intermediate steps of the

derivations are omitted in the interest of brevity.

52

First, for a land vehicle navigation system, the rotation matrix from the b -frame to the ℓ -

frame can be expressed as (Petovello 2003)

)()()( rp yxzb −−= AAAR ψℓ (3.1)

where ψ is azimuth, p is pitch, r is roll, and ψϖϖ ,, ,,, ),( rpzyxjj ==A , is a rotation

matrix about j-axis by angleϖ . Through a series of derivations with Equation (3.1), the

following is obtained (Sun et al 2008)

[ ] 0cos

0sin

cossin0

)()()( ×=

−

−+−

−

=− ΘAARA

ppr

ppr

prpr

pr xybz

ɺɺ

ɺɺɺ

ɺɺɺ

ɺ ℓ ψ

ψ

ψ (3.2)

where

cos

sin

p

r p

r pψ

= − +

Θ

ɺ

ɺ

ɺ ɺ

(3.3)

From Equation (A.1), )( b

i

b

ibbb ℓ

ℓℓɺ ΩΩRR −= . Substituting this equation and Equation (3.1)

into Equation (3.2), and through a series of derivations, the following is obtain (Sun et al

2008)

)( 11 ℓ

ℓ

ℓ

iib ωωΘ −= (3.4)

where

1 1

cos sin

( ) ( ) sin sin cos sin cos

cos sin sin cos cos

b b b

ibx ibx ibz

b b b b b

ib b ib x y iby ibx iby ibz

b b b b

ibz ibx iby ibz

r r

p r p r p p r

p r p p r

ω ω ω

ω ω ω ω

ω ω ω ω

+

= = − − = + − − + +

ω R ω A Aℓ ℓ (3.5)

53

+

−

=

−=−=ℓ

ℓ

ℓ

ℓ

ℓ

ℓ

ℓ

ℓ

ℓ

ℓ

ℓ

ℓ

ℓ

ℓ

ℓ

ℓ

ℓ

ℓ

ℓ

ℓ

zi

yixi

yixi

zi

yi

xi

zizi

ω

ψωψω

ψωψω

ω

ω

ω

ψψ cossin

sincos

)( )(1AωAω (3.6)

and where b

ibω and iωℓ

ℓ are defined in Appendix A, b

ibω is the measurement of gyros, and

1 ( ) ( )b x yp r= − −R A Aℓ . Substituting Equations (3.5) and (3.6) into Equation (3.4) yields

cos sin cos cos sin

cos sin sin cos sin cos

sin cos cos

b b n eibx ibz e

b b b

ibx iby ibz

n ee

v vp r r

M h N h

r p p r p p r

v v

M h N h

ω ω ψ ω φ ψ

ω ω ω

ψ ω φ ψ

− = + − − + + +

= + −

− − + + + +

ɺ

ɺ

tansin cos sin sin cos cos sinb b b e

ibx iby ibz e

vr p p r p p r

N h

φψ ω ω ω ω φ

− + = − + + − + + ɺ ɺ

(3.7)

where the velocity components ev , nv , meridian radius M , prime vertical radius N ,

height h , latitude ϕ , and rotation rate of the earth eω are defined in Appendix A.

Therefore, the azimuth rate equation is obtained from Equation (3.7) as

+

++−−=− ϕω

ϕωωωψ sin

tancoscossinsincossin e

eb

ibz

b

iby

b

ibxhN

vrpprpprɺɺ (3.8)

From Equation (3.7), it can be seen that since b

ibxω and b

ibyω are unknown (not measured),

pitch and roll cannot be calculated. Because pitch and roll are generally small in land

vehicle applications, they can be considered as error terms and will therefore be modeled

accordingly. However, for the mechanization equations, they are assumed to be zero,

that is, 0 , 0 == rp . With this assumption, the azimuth rate equation and attitude

direction cosine matrix can be obtained from Equations (3.8) and (3.1), respectively, and

written as

54

tansin

( )

b eibz e

b z

v

N h

φψ ω ω φ

ψ

= − + +

+ =R Aℓ

ɺ (3.9)

Substituting Equation (3.9) into Equation (A.1), the navigation equations of RIMU

(3A1G) are obtained, see Equation (3.19).

3.1.2 Derivation of Error Model of RIMU

The corresponding error model of the RIMU (3A1G) needs to be derived from the full

IMU error equations given in Equation (A.6) (see Appendix A). In order to facilitate the

derivations, in the following, fW and ωW of Equation (A.6) (i.e. accelerometer and gyro

measurement noise, respectively) are ignored in the derivation. It does not affect the

derivation, and the two terms can be added back after the derivation. The position error

model and accelerometer bias model of the 3A1G are the same as those of Equation

(A.6), but the other error state models (i.e., velocity, attitude and gyro biases) are derived

in below.

First, the attitude error model is derived. Recall that in Equation (3.9) the pitch and roll

are assumed to be zero and considered as error terms. However, for land applications, the

pitch and roll are primarily determined by the local terrain and since this terrain can be

expressed as a first-order Gaussian Markov (GM) process (Kuchar 2001), the pitch and

roll can also be expressed as first-order GM processes

+−=

+−=

rr

pp

wrr

wpp

α

α

ɺ

ɺ (3.10)

55

where rp αα , are the inverse of correlation time of the process, and rp ww , are driving

noise terms. The definition of the azimuth rate error is

ψψψδ ɺɺɺ −= ~ (3.11)

where ψɺ is the ‘true’ azimuth rate which can be obtained from Equation (3.8), ψɺ~ is

azimuth rate with error and can be obtained from Equation (3.9). Substituting Equation

(3.8) and the azimuth rate equation of Equation (3.9) into Equation (3.11), and through a

series of derivations and approximations, the following is obtained (Sun et al 2008)

ψψδ wd z +≈ɺ (3.12)

where ψw is the equivalent white noise corrupting the drift, zd is the drift of vertical gyro,

defined as

b b

z ibz ibzd ω ω= − ɶ (3.13)

where b

ibzω~ is the vertical gyro measurement, b

ibzω is true rotation rate about bz . The

vertical gyro drift, zd , can be modeled as a first-order GM process

dzdz wdd +−= αɺ (3.14)

where dα is the inverse of the correlation time and dw is the driving noise.

Next, in order to link the attitude error of the RIMU with the local terrain (pitch and roll),

the attitude error 1ℓε needs to be expressed as a function of the pitch, roll and azimuth

errors. Through a series of derivations, the attitude error 1ℓε is expressed as (Sun et al

2008)

56

1 δ=ε Θℓɺ (3.15)

where Θ is defined in Equation (3.3). Since p and r are small angles of only a few

degrees, and in the navigation equations, p and r are chosen as zero, Equation (3.15) is

approximated as

1

p p

r r

δ

δ

δψ δψ

− ≈ = − − −

εℓ

ɺ ɺ

ɺ ɺ ɺ

ɺ ɺ

(3.16)

Since in the navigation equations, p and r are chosen as zero, 0p p pδ = − = − ,

0r r rδ = − = − (estimate value minus truth value). Therefore, p pδ = −ɺ ɺ , r rδ = −ɺ ɺ (i.e.

Equation (3.16) is satisfied). From Equations (3.10), (3.12) and (3.16), the attitude error

model can be written as

+

−

+

−

−

=

ψε

ε

ε

α

α

ε

ε

ε

w

w

w

d r

p

z

z

y

x

r

p

z

y

x

1

0

0

000

00

00

1

1

1

1

1

1

ℓ

ℓ

ℓ

ℓ

ℓ

ℓ

ɺ

ɺ

ɺ

(3.17)

The velocity error model can be obtained from Equation (A.6) as

bRεFRvBbRεFvBvℓℓℓℓ

ℓ

ℓℓℓℓℓℓɺ

bb +−=+−= 11

1δδδ (3.18)

Finally, substituting Equations (3.14), (3.17) and (3.18) into Equation (A.6), the error

model of RIMU (3A1G), see Equation (3.20), is obtained.

As mentioned above, in the LTP, the pitch and roll are modelled as first-order GM

processes. Although the first-order GM models match the practical local terrain well, the

error between the GM model and the actual local terrain still exists. This is because the

GM model is obtained from the statistics of large local terrain data set (Kuchar 2001). For

57

a section of finite-length road, the local terrain and the GM model may be different. In

this light, the error in local terrain modeling needs to be considered in the velocity error

model, i.e. Equation (3.18). From Equation (3.18), it can be seen that if the error in local

terrain modeling is 1δεℓ , it will produce an acceleration error 1 1

1 δR F εℓ ℓ ℓ

ℓ in δ v

ℓɺ , and

1 1

1 1 10T

r p prg gδ η η ≈ − − = R F ε R R ηℓ ℓ ℓ ℓ ℓ

ℓ ℓ ℓ, if 1 0

T

p r prδ η η = = ε ηℓ , where g is

gravity acceleration, pη and rη are pitch and roll modeling errors (caused by error in the

local terrain model, not by errors in estimating pδ and rδ ), respectively. Since

1 prR ηℓ

ℓlooks like an accelerometer bias or accelerometer measurement noise in Equation

(3.18) (if accelerometer measurement noises are considered, as shown in Equation (A.6),

and 1 1( )b

pr b pr=R η R R ηℓ ℓ

ℓ ℓ), in practice, this term can be combined into the accelerometer

white noise or the accelerometer bias. This is implemented by setting the covariance of

either the accelerometer white noise or the accelerometer bias to be slightly larger than its

actual value in the Kalman filter of a GPS/RIMU (especially for high accuracy

accelerometers). This can be seen as a compensation for the term 1 prR ηℓ

ℓwhich does not

appear in the error model of the RIMU. Doing so obviates the need to model prη , which

is fortunate since modeling for prη is difficult.

3.1.3 Equation Summary for 3A1G Configuration

The navigation equations, which are summarized from Equations (3.9) and (A.1), are as

follows

58

1

(2 )

tansin

( )

b

b ie e

b eibz e

b z

v

N h

φψ ω ω φ

ψ

− =

= − + +

= − + + + =

r D v

v R f Ω Ω v g

R A

ℓ ℓ

ℓ ℓ ℓ ℓ ℓ ℓ

ℓ

ℓ

ɺ

ɺ

ɺ (3.19)

where ψ is azimuth, )(ψzA is a rotation matrix about the z -axis by angleψ , and the

other variables and parameters are defined in Equations (3.9) and (A.1).

The corresponding error model, which is summarized from Equations (3.14), (3.17),

(3.18), and (A.6), is given by

13 13 3 3 3 3 1 3 3

1

3 3 1 3 1

1 1

3 3 3 3 3 3

1 3 1 3 1 3 1 3

3 3 3 3 3 3 3 1

b fb

d

ddz z

b

wd d

εε

δ δ

δ δ

α

−×× × × ×

× ×

× × ×

× × × ×

× × × ×

− = +

− −

00 D 0 0 0r r

R W0 B R F 0 Rv v

W0 0 Γ M 0ε ε

0 0 0 0

W0 0 0 0 Λb b

ℓ ℓ

ℓℓ ℓ ℓℓ ℓ

ℓ

ℓ ℓ

ɺ

ɺ

ɺ

ɺ

ɺ

(3.20)

where )(1 ψzAR =ℓ

ℓ, [ ]×= b

b fRF11 ℓℓ is a skew-symmetric matrix.

)0( rpdiag ααε −−=Γ , [ ]Td 100 −=M . Other variables and parameters are

defined in Equations (3.14), (3.17), (3.18), and (A.6).

3.1.4 Mechanization and Error Model of 2A1G Configuration

In this configuration, there are two horizontal accelerometers and one vertical gyro. The

acceleration and rotation information are therefore both incomplete, and the difference

between the 2A1G and 3A1G configurations is that in the former the vertical specific

force needs to be calculated.

59

Suppose the specific force can be expressed as [ ]T

une fgff δ+=ℓf in the local level

frame ( ℓ -frame), where g is gravity acceleration and ufδ is the vehicle’s true vertical

acceleration, ef and nf are specific forces in east and north, respectively. So in the body

frame (b-frame), specific force can be expressed as

Tb b b b b

x y zf f f = = f R f ℓ

ℓ (3.21)

Substituting bRℓ

, which can be obtained from Equation (3.1) with ( )b T

b=R Rℓ

ℓ, into

Equation (3.21) yields (Sun et al 2008)

rpfrpg

rprfrprff

u

ne

b

z

coscoscoscos

)cossincossinsin()cossinsinsincos(

δ

ψψψψ

++

−−+−= (3.22)

Since pitch, roll and ufδ are generally small (most of ufδ is less than 10/g and behaves

like noise), and in most vehicle navigation applications, ef and nf are much less than g ,

Equation (3.22) can be simplified as

rpgf b

z coscos ≈ (3.23)

Since the approximation error of Equation (3.23) is related to the local terrain, which is

expressed as a first-order GM process, the approximation error of Equation (3.23) is

therefore also expressed as a first-order GM process

fzzzz wbb +−= βɺ (3.24)

where zβ is the inverse of the correlation time of zb , fzw is the driving noise of zb .

Substituting Equation (3.23) into Equation (3.19), the navigation equations of 2A1G are

obtained, which have the same form as Equation (3.19) except that the vertical

60

acceleration in the body frame needs to be calculated with Equation (3.23) and the best

available estimates of the pitch p and roll r . Substituting Equation (3.24) into Equation

(3.20), the corresponding error model of 2A1G is obtained, which has the same form as

Equation (3.20) except for the vertical accelerometer bias term. Although the vertical

accelerometer bias has the same expression in both 3A1G and 2A1G, it has different

parameters or meanings in the two error models. In 3A1G, the bias model comes from the

actual vertical accelerometer, but in 2A1G it comes from the vertical acceleration

calculation error, which is related to the local terrain.

3.2 Comparison of Three Types of RIMU Mechanizations and Corresponding Error

Models

In the following, the term “RIMU mechanization and corresponding error model” is

simplified as “RIMU M&E equations”, where “M&E” means “mechanization and error”.

In addition to the above LTP type of RIMU M&E equations obtained with the LTP

method, there are other two types of RIMU M&E equations that are often used in practice.

These two types of RIMU M&E equations are called dead reckoning (DR) type and full

dimension (FD) type, respectively. The DR type is given in Xing & Gebre-Egziabher

(2009), Iqbal et al (2008), and Rogers (1999), and the FD type is given in Nui et al

(2007b). Since the main differences among the three types of M&E equations are in their

error models, the comparison among them is made primarily in this context. In order to

compare all three types of RIMU M&E equations (LTP, DR and FD), the DR and FD

type mechanizations and their corresponding error models are introduced first. In the

following, for the sake of convenience, the three types of RIMU M&E equations are

termed as LTP, DR, and FD M&E equations.

61

3.2.1 DR Mechanization and Corresponding Error Model

The DR method is generally used in 2A1G case or an odometer plus a vertical gyro case

(Xing & Gebre-Egziabher 2009; Iqbal et al 2008; Farrell & Barth 1999). In order to

completely compare the three types of M&E equations, not only the DR M&E equations

for 2A1G but also those for 3A1G are given in here. In the DR method, the pitch and roll

are assumed to be zero in both the mechanization equations and error model of an RIMU

(Xing & Gebre-Egziabher 2009; Iqbal et al 2008). Accordingly, the DR mechanization

equations used in this chapter are obtained as Equation (3.19), and for 3A1G, the vertical

specific force b

zf comes from the vertical accelerometer measurement; for 2A1G,

b

zf g= . These DR mechanization equations are slightly different from the traditional

mechanization equations of a DR system (Xing & Gebre-Egziabher 2009; Farrell & Barth

1999). For example, Equation (3.19) will become the traditional mechanization equations

of a DR system (2A1G) (Xing & Gebre-Egziabher 2009; Farrell & Barth 1999) if the

following three modifications for Equation (3.19) are made: first the terms

(2 )ie e+Ω Ω vℓ ℓ ℓ

ℓ, gℓ , and

tansine

e

v

N h

φω φ

+

+ are removed from Equation (3.19); second,

the matrix 1−D of Equation (3.19) is assumed to be a unit matrix; and finally the height,

vertical velocity, and acceleration are not considered in Equation (3.19).

The corresponding error model for 2A1G can be obtained from Equation (3.20). From

this equation, velocity error can be written as

1 1

1( )b

b b fδ δ= + − +v B v R b R F ε R Wℓ ℓ ℓ ℓ ℓ ℓ

ℓɺ (3.25)

62

Integrating Equation (3.16) and assuming that both sides have same initial values yield

1

0

0

0

pr

p p

r r ψ

δψ δψ

− − = − = − + = + − −

ε θ θℓ (3.26)

where [ ]0T

pr p r= − −θ , [ ]0 0T

ψ δψ= −θ . Substituting Equation (3.26) into the term

1 1

1

bR F ε

ℓ ℓ

ℓof Equation (3.25) yields

1 1 1

1

b b b b

pr xyδψ= = +R F ε F ε F θ Fℓ ℓ ℓ

ℓ (3.27)

where b b = × F f , T

b b b b

x y zf f f = f , and 0T

b b b

xy y xf f = − F . Substituting Equation

(3.27) into Equation (3.25) yields

b

b xy b com b fδ δ δψ= − + +v B v R F R b R Wℓ ℓ ℓ ℓ ℓɺ (3.28)

where comb is

b

com pr= −b b F θ . (3.29)

From Equation (3.29), it can be seen that the effect of the local terrain (pitch and roll) on

the velocity error is combined into the accelerometer bias. The new accelerometer bias

comb effectively contains both the true accelerometer bias and the effect of the local

terrain.

Finally, the corresponding error model for 2A1G is expressed as

63

13 13 3 3 1 3 1 3 3

3 3 3 1

1 3 1 3 1 1 1 3

1 3 1 3 1 1 1 3

3 3 3 3 3 1 3 1

1

bb fb xy b

dz d z

bcomcom com com

w

wd d

ψ

δ δ

δ δ

δψ δψ

α

−×× × × ×

× ×

× × × ×

× × × ×

× × × ×

− = +

− −

0r 0 D 0 0 0 r

R Wv 0 B R F 0 R v

0 0 0 0

0 0 0 0

Wb 0 0 0 0 Λ b

ℓ ℓ

ℓℓ ℓ ℓ ℓ

ɺ

ɺ

ɺ

ɺ

ɺ

(3.30)

where ( )com comx comy comzdiag β β β=Λ are the inverse of the correlation times of comb ,

and bcomW are driving noises of comb (assumed to be first-order GM processes).

Combining the effect of the local terrain into the accelerometer bias term is reasonable

because both of them can be expressed as a first-order GM process, but with different

correlation times. Because of the different correlation time, modeling for the composite

bias is difficult. Combining the local terrain into the accelerometer bias means that in the

Kalman filter of a GPS/RIMU, if the variance of the accelerometer bias is chosen as

larger than its actual value, the effect of the local terrain will likely be absorbed in the

accelerometer bias term. In Equation (3.30), the position error, velocity error and

accelerometer bias each have three dimensions (3D). This is because the vertical errors

(position, velocity, and acceleration) caused by the local terrain is considered in the error

model. For 3A1G, its corresponding error mode has the same form as Equation (3.30)

except that its vertical accelerometer bias model only comes from the vertical

accelerometer error characteristics with no contribution from the local terrain.

3.2.2 FD Mechanization and Corresponding Error Model

The FD type of mechanization equations and corresponding error model are given in Nui

et al (2007b), and have the same form as those of a full IMU (see Appendix A), as the

name implies. The differences in mechanization between an RIMU and a full IMU are

their input signals. In a full IMU, all the input signals for the mechanization equations are

64

real signals obtained from actual accelerometers and gyros. But in an RIMU, such as in a

2A1G, the measurements of the rotation rates around two horizontal axes are supposed to

be zero, and the measurement of the specific force in vertical direction is supposed to be

gravity acceleration (i.e. g ). This is because there are no horizontal gyros and vertical

accelerometer. In their corresponding error models, all the senor error models of the full

IMU comes from actual sensors, whereas some senor error models of the 2A1G are

fictitious. For example, the error models for two horizontal gyros and one vertical

accelerometer are chosen as a Gaussian white noise with large variance value (Nui et al

2007b). For 3A1G case, both the vertical specific force measurement and vertical

accelerometer error model come from the actual vertical accelerometer.

3.2.3 Comparison of Three Types of Mechanizations and Corresponding Error Models

All three sets of mechanization equations come from the full IMU equations based on the

assumption that the pitch and roll are zero. And the DR and LTP mechanization

equations are the simplified form of FD equations. Correspondingly, all three sets of

equations are very similar. The only difference between the LTP mechanization and the

other two mechanizations is that in LTP mechanization, the vertical specific force

is ˆ ˆ ˆ cos cosb

zf g p r= , as shown in Equation (3.23), whereas in the DR and FD

equations, ˆ b

zf g= .

In contrast, the three error models show some important differences. The main

characteristics for each are summarized below.

65

The key characteristics of the DR error model are:

• It has 11 states (three position errors, three velocity errors, one azimuth error, one

vertical gyro drift, and three accelerometer biases).

• The effect of the local terrain (pitch and roll) on velocity error is combined into

horizontal accelerometer biases.

• As a result of the previous point, modeling of the composite bias is difficult since

the actual accelerometer bias and the local terrain have different correlation times.

• It has strong constraints on horizontal attitude error (i.e. pitch and roll are chosen

as zero). This may degrade the performance of a GPS/RIMU when the composite

bias cannot be modeled properly (since its modeling is difficult, as discussed

above).

The key characteristics of the LTP error model are:

• It has 13 states (three position errors, three velocity errors, three attitude errors,

one vertical gyro drift, and three accelerometer biases).

• Local terrain model (pitch and roll) is introduced into the RIMU error model and

used to compensate the velocity error, thus improving performance.

• Every variable in the error model has a clear concept. In particular, from the LTP

model, it can be seen clearly that the horizontal attitude error is roughly equal to

the pitch and roll, and the azimuth error is affected by the local terrain.

• Factors related to unmodeled errors/noises are clear, and it is relatively easy to

determine the covariance of process noise. In particular, the noise in the azimuth

error equation is related to the local terrain, as shown in Equation (3.12), and its

variance calculation needs to consider the local terrain factor.

66

• It has reasonable constraints (i.e. with local terrain) on horizontal attitude error

which is roughly equal to the pitch and roll.

The key characteristics of the FD error model are:

• It has 15 states (three position errors, three velocity errors, three attitude errors,

three gyro drifts, and three accelerometer biases).

• The modeling of horizontal attitude errors is not reasonable. Since the rotation

rates around two horizontal axes are modeled as white noise with large variances,

the horizontal attitude error is effectively a random walk, not a first-order GM

process. Owing to the fact that the variance of a random walk process can become

infinite as time approaches infinity (Gelb 1974), the variances of the horizontal

attitude errors can become very large with time increasing (in a quarter of a

Schuler period). This means that although the actual horizontal attitude error may

be only few degrees (for pitch and roll), the RIMU error model is able to give

unreasonably large estimates of pitch and roll. From this point, the modeling for

horizontal attitude error is considered unreasonable.

• It has no constraints (from local terrain or environment) on horizontal attitude

errors, just like in a full IMU (for a full IMU, generally its horizontal attitude error

reaches the maximum value in a quarter of a Schuler period (Farrell & Barth

1999)). And the variances of the measurement noises for horizontal rotation rates

are chosen to be on the order of a few deg/s. This may degrade the performance of

a GPS/RIMU.

67

3.3 Field Vehicle Test

In order to evaluate the above algorithms, a field vehicle test was conducted in a

suburban area of Calgary, Alberta in October 2007. Data from the field test was collected

and stored for post-mission processing. Details regarding the test are given below.

3.3.1 Test Setup, Route Selection, and Data Collection

Two grades of IMUs were used during the test: a tactical-grade IMU (Honeywell

HG1700 AG11, or simply “HG1700”) and a micro-electro-mechanical system (MEMS)-

grade IMU (Crista IMU). For the HG1700 IMU, the gyro drift is 1 deg/h and the

accelerometer bias is 1 milli-g (NovAtel 2009; Petovello 2003). For the Crista IMU, the

gyro turn on drift is 2000-5000 deg/h, noise is 200-300 deg/h/ Hz , and the

accelerometer turn on bias and noise are 0.3-0.5 m/s2 and 0.003-0.004 g/ Hz ,

respectively (CCT 2006; Godha 2006). Data from both IMUs was logged at a rate of

100 Hz. A NovAtel SPAN system was used to log GPS pseudorange, Doppler shift and

carrier phase measurements at a rate of 1 Hz, and for HG1700 IMU data collection

(NovAtel 2009). The Crista IMU data was time tagged using the pulse-per-second (PPS)

signal of the SPAN system’s GPS receiver, and logged to a computer. The time tagging

was implemented as follows: the PPS signal was fed into the Crista IMU which then

reported its time relative to the last PPS received. The remaining one second ambiguity is

then easily determined in post-mission. GPS intermediate frequency (IF) data was

collected with an IF data collection system consisting of a NovAtel Euro-3M card. An

oven controlled crystal oscillator (OCXO), namely a Symmetricom 1000B, was used to

68

drive the front-end. The key specifications of the OCXO are: frequency: 5 MHz, short

term stability: 121.0 10−< × , ageing per day: 101.0 10−< × (Symmetricom 2007).

Figure 3.1 is the vehicle test setup. In this figure, antenna A is for GPS IF data collection,

and antenna B is for NovAtel SPAN system. Figure 3.2 is data collection block diagram.

The left panel of this figure is for IMU and GPS raw data collection (i.e. pseudorange,

Doppler shift and carrier phase). Data collected from this block was used for reference

solution generation and RIMU mechanization research. The right panel in Figure 3.2 is

for IF data collection. The collected IF data, as well as the IMU data collected in the left

panel, is used for TLA and UT GPS/RIMU integration research (with a software GPS

receiver), which will be conducted in the following chapters. In this field test, two test

routes were selected for data collection, as shown in Figure 3.3 and Figure 3.4. Figure 3.3

is the route map of the open sky test. Figure 3.4 is the route map of the foliage test.

Actually, the environment of the open sky test is a mostly open sky with a few periods

containing foliage.

The open sky test lasted more than 20 minutes and was used for all the evaluations in this

dissertation. The foliage route lasted about 14 minutes and was only used for TLA and

UT GPS/RIMU integration research. A GPS base station was used in this test for

differential GPS (DGPS) purposes.

69

Figure 3.1 Vehicle Test Setup

Figure 3.2 Data Collection Block Diagram

70

Figure 3.3 Route Map of Open Sky Test

Figure 3.4 Route Map of Foliage Test

71

3.3.2 GPS Availability, Test Trajectory, and Reference Velocity and Attitude

Only the open sky test route is used for the RIMU mechanization work presented in this

chapter. The test route is divided into two sections: section A, whose environment is near

open sky case, and section B, whose environment contains a bit of foliage (see Figure

3.4).

The GPS availability of the open sky test is shown in Figure 3.5. From this figure, it can

be seen that in section A, in most cases, the number of tracked GPS satellites is seven or

eight. Only during a few very short periods does the number of tracked satellites (or

space vehicles – SVs) drop below seven. As such, it is concluded that there are no

naturally occurring GPS outages in section A. In contrast to section A, in section B there

are a few periods in which the number of tracked satellites is less than four, and

sometimes there are only one or two tracked satellites. In other words, there are clearly

naturally occurring GPS outages in section B. Generally, if there are only 1-3 tracked

satellites, the outage is called partial GPS outage and if there are no satellites tracked, the

outage called complete outage (Greenspan 1996).

Figure 3.5 Number of Tracked Satellites in Open Sky Test

72

The trajectory of the open sky test is shown in Figure 3.6. This figure shows that the

distance change in the south-north direction is about 5 km and about 2 km in the east-

west direction. The vertical variation is more than 100 m. Figure 3.7 shows the velocity

profile with the maximum horizontal velocity being about 25 m/s and the maximum

vertical velocity being less than 2 m/s. The reference attitude solution is shown in Figure

3.8. As expected, the azimuth solution roughly shows the orientation of the road on

which the vehicle moves. In contrast, pitch and roll generally show the slope of the road

(some vehicle-specific attitude variations are also included, but these are expected to be

small compared to the terrain variations and have relatively short duration). In the open

sky test, the pitch and roll mostly range between -3 and 3 degrees. The maximum

absolute pitch and roll are 4 and 5 degrees, respectively. The root mean square (RMS) of

the pitch and roll for the entire test are both 2.1 degrees. For section A, the pitch and roll

RMS is 2.2 degrees and 2.0 degrees, respectively. In the following discussions, the term

“local terrain” will mean pitch and roll variations only (not azimuth).

The reference solution was obtained using a differential GPS solution integrated with the

HG1700 IMU. It is assumed that the reference solution has a similar accuracy level with

that in Godha (2006), which is also generated using a DGPS/HG1700 IMU system: the

RMS of the position error of the reference solution in each direction is about 0.23 m, the

RMS of velocity accuracy 0.015 m/s in each direction, and the RMS of attitude accuracy

is 0.03 degrees in pitch and roll and 0.17 degrees in azimuth.

73

Figure 3.6 Trajectory of Open Sky Test

Figure 3.7 Reference Velocity of Open Sky Test

74

Figure 3.8 Reference Pitch, Roll and Azimuth of Open Sky Test

3.4 Data Processing

The data processing strategy is shown in Figure 3.9. A loose integration strategy was

adopted to simplify development and to assess algorithm performance. In loose

integration, an extended Kalman filter was used, as discussed in Chapter Two. In each

reduced IMU configuration (details below), both the tactical and MEMS-grade IMUs

were used. The GPS-only solution was obtained with GPS solution software

(C³NAVG²™) developed in the PLAN group at the University of Calgary. The GPS

measurement update rate of the Kalman filter was 1 Hz and the integrated solution output

rate was 10 Hz. For assessing the benefit of the GPS/RIMU with an LTP, in order to

simplify analysis, only route section A (near open sky case) was used. However, for

comparing the three types of RIMU M&E equations, in order to give a complete and fair

result, both sections A and B were used. For the sake of simplicity, only the MEMS IMU

was used for the comparison.

75

Mechanization

of

Reduced IMU

Reduced IMU

Error Model

Kalman Filter

GPS

Pseudorange

Doppler

Carrier Phase

+

-

Position

Velocity

Attitude

(3-D/10 Hz)

Vertical Gyro

Accelerometers

X

Y

Z

Reduced IMU100 Hz

1 Hz

Navigation

Solution

Software

C3NAVG2TM

1 Hz P/V 1 Hz

P/V 1 Hz

Figure 3.9 Data Processing Block Diagram

3.5 Evaluation of GPS/RIMU with an LTP

In order to verify the LTP method and to evaluate the performance of the GPS/RIMU

with an LTP, a series of tests were performed using different IMU configurations. In

particular, both RIMU configurations (3A1G and 2A1G) were tested using each grade of

IMU (tactical and MEMS), for a total of four combinations. All RIMUs are integrated

with GPS using a loose coupling strategy. In the following test, the correlation times of

the first-order GM models for pitch and roll are chosen as 10 s, and the variance of the

driving noise was chosen as 22 p pα σ (or 22 r rα σ for roll), where pα is the inverse of

correlation time and 2

pσ is the variance of the pitch.

76

3.5.1 GPS/3A1G Integration

Figure 3.10 shows the velocity and attitude errors of GPS/3A1G (HG1700) with LTP,

and Figure 3.11 is the velocity and attitude errors of GPS/3A1G (Crista) with LTP.

Statistics from the two figures are shown in Table 3.1 and it can be seen that the RMS of

the pitch and roll errors for both the HG1700 and Crista IMU are less than 0.8 degrees

from the actual local terrain values, i.e. the pitch and roll RMS are 2.2 degrees and 2.0

degrees, respectively. This means that the LTP can help to estimate the pitch and roll,

suggesting the model is valid in the GPS/3A1G case. Furthermore, comparing the

performance of the HG1700 and Crista IMUs from Table 3.1, it can be seen that the

lower grade IMU (Crista) has poorer performance. The azimuth error using the Crista is

near twice that when using the HG1700. Azimuth accuracy is affected by the grade of the

IMU since the azimuth is calculated from the vertical gyro measurement. But the

difference in the pitch and roll errors between the two systems is comparatively small.

With the HG1700 IMU, the pitch and roll errors are only reduced about 20% compared to

the Crista IMU. This is because the pitch and roll are affected less by the grade of the

IMU since they are computed primarily from the terrain model, which is the same in both

cases. The east and north velocity errors of the two systems are the same. This may be

explained with the following two possible reasons: the first is that because the Crista

IMU has lower pitch and roll estimation accuracy, some of its pitch and roll information

is “estimated out” in its horizontal accelerometer biases, as discussed in Sun et al (2008).

The second is that the pitch and roll estimation accuracy difference between the two

grade IMUs is too small to result in an obvious difference in east and north velocity

estimation. So the total effect from the Crista IMU on horizontal velocity is the same or

77

almost same with that from the HG1700. In contrast, the vertical velocity error is affected

by the grade of the IMU since the vertical velocity is calculated from the vertical

accelerometer measurement. Finally, from Figure 3.10 and Figure 3.11, it can be seen

that the velocity error is related to the attitude error, especially to the pitch and roll errors.

Specifically, when pitch and/or roll have large errors, the velocity error increases.

Figure 3.10 Velocity and Attitude Errors of GPS/3A1G (HG1700) with LTP

78

Figure 3.11 Velocity and Attitude Errors of GPS/3A1G (Crista) with LTP

Table 3.1 Attitude and Velocity Error Statistics of GPS/3A1G with LTP

RMS Attitude Error (Deg) RMS Velocity Error (m/s) Reduced IMU

Pitch Roll Azimuth East North Up

3A1G (HG1700) 0.57 0.58 1.38 0.08 0.08 0.05

3A1G (Crista) 0.72 0.70 2.38 0.08 0.08 0.08

3.5.2 GPS/2A1G Integration

As before, both HG1700 and Crista IMU were used in this configuration. The test results

for both IMUs are shown in Figure 3.12 and Figure 3.13, respectively. The statistics of

test results for both IMUs are listed in Table 3.2.

79

Figure 3.12 Velocity and Attitude Errors of GPS/2A1G (HG1700) with LTP

Figure 3.13 Velocity and Attitude Errors of GPS/2A1G (Crista) with LTP

Table 3.2 Attitude and Velocity Error Statistics of GPS/2A1G with LTP



2A1G (HG1700) 0.57 0.59 1.35 0.08 0.08 0.09

2A1G (Crista) 0.71 0.70 2.38 0.08 0.08 0.09

80

From the statistics, it can be seen that the pitch and roll RMS errors for both the HG1700

and Crista IMUs are reduced greatly from the local terrain variation, just as in GPS/3A1G

case. This suggests that the LTP is valid in the GPS/2A1G case as well. Further, from

Table 3.2, it can be seen that the Crista RIMU has larger attitude errors than the HG1700

RIMU, as expected. As with the 3A1G configuration, the azimuth error of the GPS/2A1G

(Crista) is almost twice that of the GPS/2A1G (HG1700) and the pitch and roll errors of

the HG1700 IMU are only reduced about 20% compared to the Crista IMU for the same

reasons as before. From Table 3.2, it also can be seen that the two grades of IMUs have

the same velocity error. For the east and north velocity errors, the reason for this result is

the same as with the 3A1G configuration. But for the vertical velocity error, the reason is

that the vertical accelerations for both grades of IMUs are calculated from the same

formula as in the 2A1G configuration, which has no vertical accelerometer (and thus is

not a function of IMU quality). Finally, from Figure 3.12 and Figure 3.13, it can be seen

that velocity error is related to attitude error, as in the 3A1G configuration.

For both 3A1G and 2A1G configurations, if the pitch and roll values of the GPS/RIMU

were fixed to zero (i.e. no LTP) (just like the DR error model when the effect of the local

terrain on the velocity error is not considered in the accelerometer bias), the velocity error

increases greatly, especially for horizontal velocities, and so does the azimuth error, as

shown in Table 3.3. Comparing this table with Table 3.1 and Table 3.2, it can be seen that

the LTP can reduce the 3D RMS velocity error by more than 80 % for both 3A1G and

2A1G. This means the LTP method is valid.

81

Through the comparison of GPS/3A1G and GPS/2A1G, two conclusions are obtained

(Sun et al 2008). First, the two configurations (3A1G and 2A1G) have almost the same

attitude result for a given grade of IMU. Second, the horizontal velocity errors are not

affected by the configuration or the vertical accelerometer, only the vertical velocity error

is affected by the configuration. When the tradeoff between cost and performance is

considered, the 2A1G configuration may be a reasonable choice for vehicular

applications.

Table 3.3 Attitude and Velocity Error Statistics of GPS/RIMU (Crista) without LTP



3A1G (Crista) 2.17 1.99 15.90 0.69 0.74 0.13

2A1G (Crista) 2.17 1.99 15.90 0.69 0.74 0.09

3.5.3 GPS Outage Test

To assess the performance of the GPS/RIMU system with an LTP during a GPS outage, a

series of GPS outage tests were conducted. In the outage tests, both grades of IMUs and

both RIMU configurations were used. Ten 30 s long GPS outages (complete) were

simulated in the data by artificially omitting the satellites during post-mission processing.

These outages were carefully selected to represent varying vehicle dynamics, including

stops, periods of acceleration/deceleration, and constant velocity.

During an outage, the “GPS/RIMU with LTP” approach assumes that the pitch and roll

estimates are almost constant values (they vary slightly because of the first-order GM

process model with long correlation time (500 s) used). In contrast, the “GPS/RIMU

82

without an LTP” assumes that the pitch and roll estimates are zero during GPS outage. In

the tests, the RMS of position or velocity error is calculated from the following equation

( )∑=

−=10

1

2)()(

10

1)(

j

iriji txtxtRMS (3.31)

where it is GPS outage duration (from 0 to 30 s), )( ij tx is the output (position or velocity)

of reduced IMU in the jth GPS outage at it and )( ir tx is the reference solution at it .

Results of the MEMS IMU are shown in Figure 3.14 and Figure 3.15 (the tactical IMU

has similar results and are therefore not shown). Figure 3.14 shows the horizontal

position and velocity errors of two configurations of GPS/RIMU (Crista) with and

without an LTP. Figure 3.15 shows the corresponding vertical position and velocity

errors. For the vertical position, since the GPS/RIMU has a larger bias (about 4 m) from

standalone GPS compared to the reference solution, in order to facilitate the analysis this

bias is removed when calculating the RMS of the vertical position error with Equation

(3.31). From Figure 3.14 and Figure 3.15, it can be seen that the position and velocity

RMS errors are a function of time since the last GPS measurement. Furthermore, Figure

3.14 shows that for the Crista IMU under both configurations, both the horizontal

position and velocity error of the GPS/RIMU with an LTP are about half those of the

GPS/RIMU without an LTP, respectively. Figure 3.15 shows that both the vertical

position and velocity error are reduced only marginally or remain at the same level when

an LTP is applied.

83

In order to facilitate the comparison, the RMS position errors at the end of the GPS

outage are summarized in Table 3.4. From this table, it can be seen that, for the

GPS/RIMU without an LTP, the horizontal position errors for different grades of IMU

(HG1700 and Crista) and different RIMU configurations (3A1G and 2A1G) are almost

the same and range from 219 to 223 m. This is consistent with the earlier analysis: the

horizontal position error is mainly determined by the local terrain when there is no LTP

used during GPS outage. With an LTP, the horizontal position errors for different IMU

grades and different configurations are greatly reduced, as shown in Table 3.4. From this

table, it can be seen that with an LTP, even though the grade and configuration of the

RIMUs are different, they have similar results. Specifically, the RMSs of the horizontal

position errors for all IMUs and configurations are almost the same (from 103 to 104 m),

which is less than half the error without an LTP. Like the position errors, the horizontal

velocity errors of the GPS/ RIMU with an LTP are also reduced, and the extent of the

reduction for both position and velocity errors is similar (in terms of percentage), as

shown in Figure 3.14. For the vertical position error, Table 3.4 shows that it is reduced

only slightly, if at all, with an LTP for all the IMUs and configurations. The vertical

velocity error is similarly reduced, as shown in Figure 3.15. A detailed analysis for these

results is given in Sun et al (2008). All these test results show that the LTP is valid during

a GPS outage.

84

Figure 3.14 Horizontal Position and Velocity Error Comparison between

GPS/RIMU (Crista) with and without LTP

Figure 3.15 Vertical Position and Velocity Error Comparison between GPS/RIMU

(Crista) with and without LTP

85

Table 3.4 Horizontal and Vertical Position Error Comparison between GPS/RIMU

with and without LTP at The End of GPS Outage

RMS Position Error (m)

Horizontal Vertical

Reduced IMU

LTP No LTP LTP No LTP

3A1G (HG1700) 104 219 10 15

2A1G (HG1700) 104 223 12 12

3A1G (Crista) 103 221 11 12

2A1G (Crista) 103 220 11 12

3.5.4 Summary

With the GPS/RIMU with an LTP, a series of system configuration tests were conducted

to verify the LTP method. Following the system configuration tests, some GPS outage

tests were conducted. Based on the test results, the following conclusions are drawn:

• The LTP method is valid. With the LTP, pitch and roll were estimated, and

velocity error was reduced, especially for horizontal velocity error, it was reduced

by more than 80% for both 3A1G and 2A1G MEMS IMU compared to without

LTP.

• During GPS outages, the LTP method reduced position and velocity errors.

• During GPS outages with an LTP, the horizontal position and velocity errors of

different grade IMUs and different configurations were reduced greatly, but the

vertical position and velocity errors were reduced only slightly or remained the

same.

Given the above, 2A1G may be a better configuration when cost and performance are

considered. In conclusion, LTP is a valid attitude error model (for pitch and roll) for

RIMU, and can improve the navigation performance of a GPS/RIMU.

86

3.6 Evaluation of Three Types of RIMU M&E Equations

In order to compare the performance of the three types of mechanizations and their error

models, another series of tests were performed using only the Crista system. Specifically,

for each mechanization, two GPS/RIMU systems, namely the GPS/3A1G and GPS/2A1G,

were tested using sections A and B of the data. Consequently, there are four sets of

results for each mechanization.

In order to obtain a fair comparison between the different types of M&E equations, the

process noise parameters and covariance of each error model must be chosen properly. In

practice, this is difficult since for different error models the process noise parameters

have different interpretations, especially for those noises which are not related to or do

not come from actual sensors (e.g., related to local terrain). Herein, process noise terms

that are not related to a particular sensor are called noise without a corresponding sensor.

Furthermore, it is understood that the covariances of these noises can only be obtained

approximately because they are related to some unknown factors. For example, the

horizontal rotation rate noises are related to the rates of pitch and roll, which are hard to

be known precisely without horizontal gyros. In this light, the comparison of the different

types of M&E equations below can be seen as approximate, but still instructive. In the

following tests, the covariance of each corresponding error model is chosen as the best

value that we can have. It is obtained through several parameter selection trials.

3.6.1 Test Results with Section A (Near Open Sky Case)

First, GPS/3A1G (Crista) and GPS/2A1G (Crista) were tested using different M&E

equations with data from section A. The test results are summarized in Table 3.5 for

87

3A1G, and Table 3.6 for 2A1G. From Table 3.5, it can be seen that for the 3A1G case,

the three types of M&E equations have similar results under the near open sky case

except that the DR model can not estimate pitch and roll. For the 2A1G case, similar

conclusions can be drawn from Table 3.6. For the DR model, the test results show that

although it can not estimate pitch and roll, it still has a similar azimuth and velocity

estimation accuracy with those of the LTP model. This is because the effect of the local

terrain (pitch and roll) on the velocity error is modeled as an accelerometer bias, and is

estimated in this term. Consequently, the estimated accelerometer bias in this case is not

the true accelerometer bias but instead is a composite bias, as shown in Equation (3.29).

If the pitch and roll are not modeled in the accelerometer bias term (i.e. the accelerometer

bias model comes from the actual accelerometer), they will not be estimated, resulting in

navigation errors, as shown in Table 3.3. Comparing Table 3.3 with Table 3.5 and Table

3.6, it can be seen that when pitch and roll are not estimated in accelerometer bias, both

the azimuth and velocity estimation error will increase several times (about eight-fold for

both azimuth and horizontal velocity error), compared to the case where pitch and roll are

estimated in the accelerometer bias. For the GPS/RIMU with an LTP or FD model, its

pitch and roll information comes from the GPS solution because neither the LTP nor the

FD model can provide the local terrain information directly.

Table 3.5 Velocity and Attitude Error Statistics of GPS/3A1G (Crista) with Data

from Section A

RMS Attitude Error (Deg) RMS Velocity Error (m/s) 3A1G

Mechanization Pitch Roll Azimuth East North Up

DR N/A N/A 2.24 0.08 0.08 0.08

LTP 0.72 0.70 2.38 0.08 0.08 0.08

FD 0.61 0.74 2.48 0.08 0.09 0.07

88


from Section A



DR N/A N/A 2.24 0.08 0.08 0.09

LTP 0.71 0.70 2.38 0.08 0.08 0.09

FD 0.61 0.71 2.50 0.08 0.09 0.09

3.6.2 Test Results with Section B (Near Foliage Case)

The GPS/3A1G and GPS/2A1G configurations were tested using different types of M&E

equations with data from section B. The test results are summarized in Table 3.7 for

3A1G, and Table 3.8 for 2A1G. From Table 3.7, it can be seen that for 3A1G case, the

DR and LTP model have similar results except that the DR model can not estimate pitch

and roll. But for the FD model, both its attitude and velocity errors are much greater than

those of the LTP. In particular, the velocity error is more than twice that of the LTP. For

the 2A1G case, similar conclusions can be obtained from Table 3.8 except for the vertical

velocity error. The vertical velocity error of the FD is less than that of the LTP, which is

contrary to the result of 3A1G case. This can be explained with Figure 3.16, the vertical

velocity error of the 3A1G and 2A1G cases with the FD model and under near foliage

case. From this figure, it can be seen that a large vertical velocity error (more than 1.5

m/s) in the 3A1G case, which appears between GPS time 242800 and 242980 s, is not

present in the 2A1G case, resulting in a smaller RMS vertical velocity error.

Consequently, the RMS vertical velocity error of the 2A1G is less than that of the 3A1G,

as shown in Table 3.7 and Table 3.8, although the vertical velocity error of the 3A1G is

much less than that of the 2A1G in most cases, as shown in Figure 3.16. As a result, in

2A1G, the vertical velocity error of the FD is less than that of the LTP. The reason for

89

that with the FD error model, the large vertical velocity error in 3A1G case disappears in

2A1G case is explained as follows: in this partial outage (between GPS time 242800 and

242980 s), the tracked SVs have high elevation angles (one SV’s elevation is about 80

degrees), thus providing relatively accurate vertical information (for both position and

velocity). In this case, since the GPS/2A1G takes more vertical information from the GPS

than the GPS/3A1G does (since 2A1G has no vertical accelerometer), the large vertical

velocity error that may arise in the 2A1G case is corrected by GPS. As a result, the

GPS/2A1G does not produce any large vertical velocity error during this partial GPS

outage.

With respect to the performance comparison between the FD and LTP model (or FD and

LTP M&E equations), as concluded above, the FD model has greater attitude and

velocity error than the LTP does. This is because in the FD model, there is no constraint

on horizontal attitude errors, and the variances of the measurement noises of horizontal

rotation rates are chosen to be on the order of a few deg/s, as discussed in Section 3.2.

Consequently, the GPS/RIMU with FD model takes more navigation information from

the GPS than the system with the LTP model does, especially during partial GPS outages

(because there is no constraint in FD model). Since there are several partial GPS outages

in the near foliage test, the performance of the GPS/RIMU with FD model is degraded by

the poor GPS solutions, and the degradation is more serious than that with LTP model.

Thus the FD model has greater attitude and velocity error than the LTP does.

90


from Section B



DR N/A N/A 3.55 0.35 0.31 0.10

LTP 0.56 0.75 3.61 0.35 0.32 0.12

FD 0.81 0.99 5.30 0.80 0.80 0.26


from Section B



DR N/A N/A 3.55 0.35 0.31 0.26

LTP 0.55 0.77 3.66 0.35 0.32 0.26

FD 0.68 1.02 5.31 0.81 0.76 0.15

Figure 3.16 Vertical Velocity Error of GPS/RIMU (Crista) with FD Model in

Section B

3.6.3 Summary and Discussion

From the above test results, it can be concluded that both the DR and LTP M&E

equations (or model) perform well in both the near open sky and the near foliage case

except that the DR model cannot estimate pitch and roll. But the FD model only

performs well in the near open sky case. This is because the FD model has no constraint

91

on horizontal attitude errors. Thus the GPS/RIMU with FD model takes more navigation

information from the GPS than the system with LTP model does. When the navigation

information provided by the GPS is poor, such as in the near foliage case, the FD model

will perform poorly, i.e. with larger navigation errors (velocity and attitude) than the LTP

model.

The reason the DR model can achieve a similar performance with that of the LTP is that

the effect of the local terrain (pitch and roll) on the velocity error is modeled and

estimated in the accelerometer bias term. If the pitch and roll are not estimated in the

accelerometer bias term, both the azimuth and velocity error of the DR model increase

about eight-fold. In conclusion, compared to the LTP model, the DR model can achieve a

similar performance only if the pitch and roll are estimated in the accelerometer bias term

(a composite bias). Compared to the LTP model, the FD model can achieve a similar

performance in the near open sky case, but in the near foliage case, its performance is

much inferior to that of the LTP except for the vertical velocity in the 2A1G

configuration.

3.7 Summary

In this chapter, in order to overcome the disadvantages of RIMUs, an LTP method was

developed for RIMU to obtain a set of innovative mechanization equations and error

model. This M&E equations were compared with other two types of M&E equations

(namely DR and FD type) to investigate their relative performance. After the theoretical

research, a series of tests for GPS/RIMU were conducted with real test data to verify the

92

LTP method, and to compare the performance of the three types of M&E equations.

From these tests, the following conclusions are drawn:

• The LTP method is valid, and can improve the navigation performance of a

GPS/RIMU in both GPS available and unavailable cases.

• With an LTP, pitch and roll were estimated, and velocity error was reduced,

especially for horizontal velocity error. The 3D RMS velocity error was reduced

by more than 80% compared to without LTP case.

• 2A1G may be a better configuration for GPS/RIMU (loose) when cost and

performance are considered because with a vertical accelerometer, only the

vertical velocity error of the GPS/3A1G was reduced (about 11% for MEMS IMU).

• Compared to LTP model, the DR model can achieve a similar performance only if

the pitch and roll are estimated in the accelerometer bias term (a composite bias);

otherwise its performance is much degraded.

• Compared to the LTP model, the FD model achieved a similar performance in the

near open sky case. But in the near foliage case, its performance was much inferior.

Specifically, the 3D RMS velocity error of the GPS/3A1G (Crista) with FD model

was more than twice that of the system with the LTP model.

93

Chapter Four: TLA GPS/Reduced IMU

As discussed in Chapter One, in a tight GPS/RIMU, if the GPS receiver’s tracking loops,

especially PLLs, receive Doppler aiding from the INS obtained from the RIMU, the tight

integration system is called tight with loop aiding (TLA), i.e. TLA GPS/RIMU. In a TLA

GPS/RIMU, the noise bandwidth of the loop filters of the PLLs can be narrowed more

than in a GPS-only case (Chiou et al 2007; Petovello et al 2007; Gebre-Egziabher et al

2005; Chiou 2005). As a result, the TLA integration system offers some advantages over

GPS-only receivers including a more accurate navigation solution (especially for velocity

and thus position), improved tracking ability for high dynamics and good anti-jam

performance (Kim et al 2007; Petovello et al 2007; Chiou et al 2004; Hamm et al 2004).

Since the TLA system’s performance can be improved through narrowing the noise

bandwidth of the PLL loop filters, in order to design a TLA system, the following two

problems must be solved: how narrow can the noise bandwidth be and how much can the

system performance be improved through a Doppler aiding? To answer these two

questions, this chapter conducts a thorough assessment on the TLA GPS/RIMU system in

terms of the tracking ability of the GPS receiver and the navigation performance of the

TLA system when the loop filter noise bandwidth is adjusted.

In this chapter, first the TLA GPS/RIMU is introduced. Then the phase error of a PLL

with Doppler aiding is analyzed. Based on the phase error analysis, the formulae, which

can be used to calculate the noise bandwidth of a Doppler-aided PLL, are given. Then an

adaptive PLL loop filter – or simply an adaptive loop filter (ALF) – is designed to exploit

the greatest potential ability of the Doppler-aided PLL to improve the performance of the

94

TLA GPS/RIMU. Following the theoretical research, a test conducted with a translation

stage on the roof of a building is described first. The translation stage provides motions

for the test. Then an assessment of the potential benefits of a TLA GPS/RIMU is

performed with the translation stage test data. Further an assessment with real data

collected in the field vehicle test, as described in Chapter Three, is made to identify the

advantages of TLA system. Finally, an evaluation of the TLA GPS/RIMU with and

without ALFs is performed using the vehicle test data.

4.1 TLA GPS/RIMU

In a TLA GPS/RIMU system, the key point is how the GPS receiver uses a PLL aided

with a Doppler value calculated from the integrated navigation solution and satellite

information to improve its performance. In the TLA system, two reduced IMU

configurations are considered: 3A1G and 2A1G. Their navigation equations and error

models are given in Chapter Three. The configuration of the TLA GPS/RIMU is shown

in Figure 4.1, which gives more detailed information for the Doppler aided PLL than

does Figure 2.10 – the block diagram of TLA GPS/INS integration. It is noted in Figure

4.1 that the “Tight Integration” box contains an extended Kalman filter to integrate both

the GPS (pseudorange and Doppler shift) and IMU data. The Doppler aiding rate for the

tracking loops is equal to the IMU measurement rate, i.e. the output rate of the Kalman

filter for Doppler aiding calculation. The Doppler aiding value is calculated from

Equation (2.21). It consists of two parts: one comes from the relative motion of the

satellite with respect to the user; the other from the receiver clock drift.

95

In the TLA GPS/RIMU system, the Doppler-aided PLL can be modeled as a control

system (Alban et al 2003; Misra & Enge 2001), as shown in Figure 4.1 (right panel). In

order to facilitate the analysis of the phase errors induced by Doppler aiding error, the

true Doppler, ( )dF s , is removed from the block diagram of Doppler-aided PLL, resulting

in Figure 4.2, the block diagram of Doppler aiding error propagation. In this figure, phase

error ( ) ( ) ( )r r ds s F s sδϕ ϕ= − and Doppler aiding error )()()( sFsFsF ddcd −=∆ are

inputs and ( ) ( ) ( )ds s F s sδϕ ϕ= − is output, where ( )r sϕ is PLL input phase, ( )sϕ is

output phase, ( )dcF s is the calculated Doppler aiding. With reference to Figure 4.2, )(sN

is noise, ( ) 1pG s s= is the transfer function of the numerically controlled oscillator

(NCO), and ( )2( ) 2c n nG s s sξω ω= + is the transfer function of a second-order loop filter,

where 707.0=ξ is the damping ratio of the system and nω is the un-damped natural

frequency. Furthermore, ( ) 0r tδϕ = cannot affect the system analysis since it is assumed

that the phase error induced by system dynamics only comes from the Doppler aiding

error. From control system theory (e.g., Nise 2000; Ogata 1997), the phase error induced

by Doppler aiding error is obtained from Figure 4.2 as follows

2 2( ) ( )

2d

n n

ss F s

s sδϕ

ξω ω= ∆

+ + (4.1)

96

Figure 4.1 Configuration of TLA GPS/RIMU

Figure 4.2 Block Diagram of Doppler Aiding Error Propagation

4.2 Phase Error of Doppler-Aided PLL in TLA GPS/RIMU

In a TLA GPS/RIMU, the phase error of the Doppler-aided PLL is, to some extent,

different from the traditional PLL because Doppler aiding not only removes most user

dynamics, but also introduces some additional frequency errors. In light of this, the phase

error of Doppler-aided PLL needs to be reviewed.

4.2.1 Individual Phase Errors in Doppler-Aided PLL

There are many possible error sources in a Doppler-aided PLL. To simplify the analysis,

only the following main sources are considered herein (see Appendix B for details):

• Thermal noise (see Equation (B.1))

• Allan deviation oscillator phase jitter (see Equation (B.2))

97

• Vibration-induced oscillator phase jitter (see Equation (B.3))

• Doppler aiding error-induced phase errors

In the context of this thesis, the last error source is of primary interest and, to this end, it

is assumed that the Doppler error of the GPS/RIMU consists of the combination of a

series of piecewise first-order Gauss-Markov (GM) processes and a series of piecewise

random ramps (RRs). The reasons for doing so are twofold: first, a series of piecewise

GM and RR processes well match the Doppler error observed in most cases because of

the error characteristics of GPS/RIMU; second, using only these two types of Doppler

errors can facilitate system analysis. In GPS/RIMU integration, if the attitude error is

small and noisy, the resulting Doppler error will look like a first-order GM process,

which is produced mostly by the GPS measurement and the reduced IMU attitude error.

Otherwise, the Doppler error may contain a first-order GM process, and a random ramp

mainly induced by the reduced IMU attitude error. For example, the velocity error of the

GPS/3A1G, as shown in Figure 4.20, can be expressed as a first-order GM process. Its

correlation function can be approximately expressed as 0( )R R eβ ττ −

= , for east and north

velocity error, 3

0 1.9R e−≈ (m/s)

2, 1 / 5β ≈ (1/s), respectively; for vertical velocity error,

3

0 1.0R e−≈ (m/s)

2, 1 / 4β ≈ (1/s). The equations corresponding to the Doppler aiding-

induced phase errors of interest in this chapter are given below.

Random ramp-induced phase error is obtained from Equation (B.4), and given as

2360 RR

RR

n

Aθ

ω= (degrees) (4.2)

98

where RRA is the amplitude (slope) of the random ramp (Hz/s). First-order GM-induced

phase jitter is obtained from Equation (B.10), and given as

2 2

4 4360

22 2 2 2

T

w nGM

n n

eα

ϕ

σ γ ω γσ

ω γ ω

⋅= ⋅ + −

+ (degrees) (4.3)

where Teααγ = , and from Equation (B.8),

TeT

GMw /)1( 2ασσ −−= (4.4)

where GMσ is the standard deviation of the GM process (Hz), α is the reciprocal of the

correlation time (1/s), and T is the measurement update period of the Kalman filter for

GPS/RIMU (s).

4.2.2 Total Phase Error in Doppler-Aided PLL

Based on the above individual phase errors, the total phase error (1-sigma) of the

Doppler-aided PLL can be expressed as

2 2 2 2 2

3

RRPLL tPLL v Arx Asv GMϕ

θσ σ σ σ σ σ= + + + + + (4.5)

where tPLLσ , vσ , GMϕσ , and

RRθ are thermal noise, oscillator vibration phase jitter,

Doppler aiding GM phase jitter, and Doppler aiding RR phase error, respectively, and

Arxσ and Asvσ are respectively the receiver and satellite Allan phase jitters. tPLLσ and vσ

are defined in Equations (B.1) and (B.3), respectively. Arxσ and Asvσ are defined in

Equation (B.2). GMϕσ and RRθ are defined in Equations (4.3) and (4.2), respectively. The

RR phase error is considered as a dynamic stress error (much like the dynamic stress

error in a standard tracking loop), so it appears outside the square root in Equation (4.5)

(Ward et al 2006). In Equation (4.5), the thermal noise, oscillator Allan phase jitter,

99

oscillator vibration phase jitter, and Doppler aiding error-induced phase error are

considered to be independent. Furthermore, the Doppler aiding GM phase jitter and

Doppler aiding RR phase error are assumed to be independent to simplify the analysis of

Doppler aiding error-induced phase error. It is noted that Equation (4.5) is applicable to

any Doppler-aided PLL. Furthermore, if the parameters for calculating the total phase

error with Equation (4.5) are known, according to the PLL tracking threshold

15 (degrees)PLLσ ≤ (Ward et al 2006) (i.e. if 15 (degrees)PLLσ ≤ , the PLL is considered

to be locked), the minimum noise bandwidth of Doppler-aided PLL can be obtained with

Equation (4.5). Since PLLσ is a nonlinear function of the noise bandwidth, as shown in

Equation (4.5) (also discussed in Ward et al (2006) and Gebre-Egziabher et al (2005)),

there are no analytical solutions for the noise bandwidth. Therefore, a search strategy is

needed to obtain the minimum noise bandwidth solution. If the noise bandwidth solution

is obtained in real time and applied to the PLL loop filter simultaneously, an adaptive

loop filter with varying noise bandwidth can be obtained.

4.3 Adaptive PLL Loop Filter for TLA GPS/RIMU

In general, each PLL tracking loop of a GPS receiver is controlled by a loop filter whose

order and noise bandwidth impact the performance of the tracking loop (Ward et al

2006). In a TLA GPS/RIMU integration, the noise bandwidth can be narrowed more than

in a GPS-only case resulting in improved noise suppression within the receiver (Petovello

et al 2007, Alban et al 2003). However, the loop noise bandwidth cannot be made

arbitrarily small and is limited by the navigation solution error of the GPS/RIMU and the

receiver’s oscillator errors (Gebre-Egziabher et al 2005), as shown in Equation (4.5).

Since the navigation solution error varies (see Chapter Three), the loop filter with a

100

constant noise bandwidth cannot optimally accommodate signal dynamics resulting from

the navigation solution. In this light, an adaptive PLL loop filter capable of adjusting its

bandwidth would be beneficial, as it would still be able to accommodate large navigation

errors, but would provide improved performance during periods when the navigation

solution errors are small.

However, to date, there has not been a method or formula to calculate the noise

bandwidth according to the navigation performance of a TLA GPS/RIMU. How to

choose the noise bandwidth is thus an open problem that is addressed herein. In order to

solve this problem, based on the phase error analysis given in Gebre-Egziabher et al

(2005) and Chiou (2005), the phase error sources for the GPS/RIMU are analyzed, and

formulae for calculating these phase errors are derived in Appendix B. Following the

phase error analysis, a novel adaptive PLL loop filter is developed. Generally, it is

straightforward to manipulate the loop bandwidth because it is easier to maintain a stable

loop during transitions compared to manipulating the loop order (Lee et al 2007; Hsieh &

Sobelman 2005; Mao et al 2004). For adjusting the noise bandwidth of an adaptive loop

filter, some information can be used to calculate the noise bandwidth, such as carrier to

noise power density (C/N0), elevation angle of the satellite, and the vehicle’s acceleration

(Lee et al 2007). Actually in Lee et al (2007), the noise bandwidth only takes on two

values (i.e. a wide noise bandwidth and a narrow noise bandwidth) according to the C/N0,

elevation, and acceleration. This method is quite coarse.

101

4.3.1 Evaluation of Doppler Aiding Uncertainty for TLA GPS/RIMU

From Equation (4.5) (i.e. the total phase error of Doppler-aided PLL), it can be seen that

the Doppler aiding error (as computed from the integrated navigation solution) affects the

total phase error of the PLL. Therefore, in order to calculate the total phase error and

design the ALF, the Doppler aiding uncertainty resulting from the navigation uncertainty

is derived first below.

The Doppler aiding error uncertainty can be calculated from the GPS/RIMU navigation

solution uncertainty. Specifically, the navigation solution uncertainty – as obtained from

the integrated navigation solution Kalman filter predicted covariance matrix, ( )−P – is

used to compute the Doppler aiding uncertainty, which is then used to adapt the tracking

loop. In order to link )(−P with real GPS measurements, an adaptive algorithm for

calculating )(−P is used as follows (Mohamed & Schwarz 1999)

11ˆ)()( −− ++=− kkk QPP (4.6)

where the process noise covariance is adapted using

T

k

T

kkkkk 11111 )1(ˆˆ−−−−− ⋅−+⋅= KυυKQQ ηη (4.7)

where 10 <<η is a weighting factor, 1−kK is the Kalman filter gain,

)(1111 −−= −−−− kkkk XHzυ is the innovation sequence of the Kalman filter, where 1−kz is

the measurement, 1−kH is measurement matrix, )(1 −−kX is the predicted state vector. The

initial value 0Q is chosen as the best estimate of Q based on system and error

characteristics.

102

Suppose the diagonal elements (sub-matrices in diagonal) of the ( )−P are ( )rδ −P ,

( )vδ −P , ( )ε −P , ( )d −P , and ( )b −P , where ( ) ( ) ( ( ))( ( ))T

g E − = − − P g g , ( )δ −r denotes

predicted position error, ( )δ −v denotes predicted velocity error, ( )−ε denotes predicted

attitude error, ( )−d denotes predicted gyro drift, and ( )−b denotes predicted

accelerometer bias. Once )(−P is obtained, the velocity error sub-matrix ( )vδ −P can be

extracted. ( )vδ −P is then used to compute the Doppler aiding uncertainty caused by the

velocity error of the TLA GPS/RIMU as

2

1

2 ˆ)(ˆ

λσ δ κPκ −

= v

T

fd (4.8)

where 11 / Lfc=λ , c is the speed of light,

1Lf is the transmitted frequency, and κ is the

unit vector from the user to SV. The position error is assumed small enough that κκ ≈ˆ

(this is reasonable for position errors less than several hundred metres). Similar to the

above, the variance of the Doppler rate caused by the integrated system’s acceleration

error is given as

2

1

2 ˆ)(ˆ

λσ δ κPκ −

= a

T

afd (4.9)

where )()( −=− va ɺδδ is the predicted acceleration error. The calculation of )(−aδP is

derived in Equation (4.10). In a reduced IMU, since bRεFRv ℓℓℓℓ

ℓɺ

b+−≈ 11

1δ (see Equation

(3.18)), then

T

bbb

TT

a ))(()())(()( 1

11

1

ℓℓℓ

ℓ

ℓℓℓ

ℓRPRRFPFRP −+−=− εδ (4.10)

103

where 1ℓε is attitude error in the 1ℓ -frame, [ ]×= b

b fRF 11 ℓℓ is a skew-symmetric matrix,

where bf is specific force in the b-frame, ℓ

ℓ1R is a rotation matrix, and b is the

accelerometer bias. The ℓ -frame is the local level frame (east, north, and up) and the 1ℓ -

frame is an alternative level frame (see Section 2.1 for details). Once )(−P is known, the

sub-matrices )(−εP and )(−bP can be extracted and )(−aδP is calculated using Equation

(4.10).

For one SV channel, once 2

fdσ is obtained, the Doppler aiding-induced phase jitter can be

calculated. Recall that the Doppler error of TLA GPS/RIMU is assumed to be the

combination of a series of piecewise first-order GM processes and a series of piecewise

random ramps. The objective therefore is to relate the values of 2

fdσ and afdσ from above

to the variance of the first-order GM process in Equation (4.4) (i.e., 2

GMσ ) and to the

amplitude of the RR error in Equation (4.2) (i.e., RRA ). It is pointed out that there is no

rigorous method to accomplish this. However, as will be shown later, the following

approximations work well in practice

2 2

GM fdσ σ≈ (4.11)

3RR afdA σ≈ (4.12)

The first equation states the GM variance for each satellite is approximately equal to the

Doppler uncertainty for that satellite. The second equation considers the 3σ (>99%)

Doppler rate uncertainty to be a good upper bound of the slope of the random ramp.

104

It is noted that for a GPS/RIMU, the velocity error has different characteristics in the

horizontal and vertical directions (see Section 3.5 for details). The Doppler aiding

uncertainty caused by the velocity error of the TLA GPS/RIMU is therefore separated

accordingly. The horizontal and vertical Doppler aiding uncertainties are assumed to have

different correlation times for the GM process. In this context, the Doppler aiding

uncertainty caused by horizontal velocity error is given as

2

1

2ˆ)(ˆ

λσ δ κPκ −

=vxy

T

fdxy (4.13)

where [ ]Tyxxy vv 0)()()( −−=− δδδv . Similarly, the Doppler aiding uncertainty caused

by vertical velocity error is

2

1

2 ˆ)(ˆ

λσ δ κPκ −

= vz

T

fdz (4.14)

where [ ]T

zz v )(00)( −=− δδv . Once2

fdxyσ and 2

fdzσ are obtained, 2

GMxyσ and 2

GMzσ −

the variances of the first-order GM − can be calculated using Equation (4.11). Then,

using Equation (4.3), the phase jitters GMxyϕσ and

GMzϕσ induced by 2

fdxyσ and 2

fdzσ

respectively can be calculated. Finally, the total phase jitter caused by GM Doppler

aiding uncertainty is

2 2 2

GM GMxy GMzϕ ϕ ϕσ σ σ= + (4.15)

Having developed a means of approximating the different error components in a TLA

GPS/RIMU system, the following section looks at how the ALF is designed.

105

4.3.2 Adaptive PLL Loop Filter Design

As suggested in Chiou (2005), a second-order loop filter is used in the Doppler-aided

PLL of the TLA GPS/RIMU. A second-order loop filter has two parameters; the damping

ratio, ξ , and the natural frequency, nω . When 707.0=ξ , the system has not only the

almost minimum coefficient between noise bandwidth and natural frequency, i.e. noise

bandwidth nnB ω53.0= (the minimum coefficient is 0.5 (Stensby 1997)), but also an

optimal dynamic performance of the system, i.e. the system’s overshoot and settling time

are both small (Best 2007; Ogata 1997) . As a result, in the adaptive loop filter developed

here, only the noise bandwidth is adjusted.

In a TLA GPS/RIMU, the total phase error variation with the noise bandwidth is shown

in Figure 4.3. From this figure, it can be seen that when the GPS signal is strong (C/N0 =

45 dB-Hz), the total phase error will increase with decreasing noise bandwidth. When the

C/N0 is decreased to 30 dB-Hz, the total phase error still has a similar variation tendency

as in the strong signal case. In contrast, when the C/N0 is further decreased to 20 dB-Hz

(i.e. the GPS signal is very weak), the total phase error has a minimum value at a

particular noise bandwidth value, and it is always greater than 15 degrees. For the results

shown, the predetection integration time (PIT) was 20 ms, and similar conclusions can be

drawn for other PIT values.

106

Figure 4.3 Total PLL Phase Error Variation with Noise Bandwidth

Given the above, the selection of the tracking loop’s noise bandwidth was made

according to two criteria:

• For a strong signal (i.e. a signal for which the total phase error is less than 15 deg

at a noise bandwidth of 10 Hz – see the middle and the bottom curves in Figure

4.3) the lowest bandwidth for which the total PLL error is less than 15 deg is

selected. The motivation here is that as long as the total PLL error meets this

requirement, the tracking loop is considered to be tracking the phase of the signal

(Ward et al 2006) and so by minimizing the loop bandwidth the overall Doppler

error is also minimized.

• For a weak signal (i.e. a signal for which the total phase error exceeds 15 deg at a

noise bandwidth of 10 Hz – see the top curves in Figure 4.3) the bandwidth that

107

minimizes the total PLL error is selected to allow the PLL the maximum

possibility to be locked.

Based on the relationship between total phase error and noise bandwidth, the noise

bandwidth search flowchart is given in Figure 4.4.

Figure 4.4 Noise Bandwidth Search Flowchart

4.4 Test Description

To explore the advantages of TLA GPS/RIMU and to verify the ALF algorithm, two

kinds of tests were conducted to collect data for post-mission processing: one is

translation stage test, which is only used for TLA system research; the other is field

vehicle test, which was described in Chapter Three and will be used for both TLA system

and the ALF research.

108

4.4.1 Translation Stage Test

The translation stage test was conducted on the roof of the engineering building at the

University of Calgary in March 2007. The roof is a relatively benign signal environment

(i.e., little multipath and few obstructions). So it can provide an ideal or near ideal

environment to investigate the advantages of TLA GPS/RIMU. In this test, motion was

provided with a precise translation stage (Anorad, relative precision at the micrometre

level) that was approximately level. The precise motion allowed for a controlled motion

profile to be generated and used as a reference solution. The maximum acceleration

generated for the test was 1.9 m/s2 (this is the most the table could generate). A picture of

the test setup is shown in Figure 4.5. A Honeywell HG1700 IMU (described in Section

3.3) and a GPS antenna were installed on the translation stage. The HG1700 IMU is part

of the NovAtel SPAN system (see Section 3.3 for details). For time tagging purposes, the

SPAN system’s receiver was fed data from a separate (static) antenna. The RF signal

from the antenna mounted to the translation stage was passed through a variable signal

attenuator and then to an IF data collection system consisting of a NovAtel Euro-3M card

(see Section 3.3 for details). The attenuator attenuated the signal at a rate of 1 dB/s for 40

s before the end of the test. An OCXO, namely a Symmetricom 1000B (see Section 3.3

for details), was used to drive the front-end (for more details, see Petovello et al (2007)).

109

Figure 4.5 Translation Stage Test Setup

A skyplot of the translation stage test is shown in Figure 4.6. From this figure, it can be

seen that there were eight SVs in view with an elevation greater than 5 degrees. The

reference velocity of the stage motion in forward-backward direction is shown in Figure

4.7. It is obtained from the INS plus zero velocity update (ZUPT) since GPS

measurements (i.e. pseudorange and Doppler shift) were unavailable in this test. The

velocities of the stage in both vertical and lateral direction were very small (less than 0.02

m/s). Furthermore, all the dynamics were in forward-backward direction, which was

approximately in a north/south direction.

110

Figure 4.6 GPS Satellite Skyplot of Translation Stage Test

Figure 4.7 Reference Velocity of Stage Motion in Forward Direction

4.4.2 Field Vehicle Test

The field vehicle test was described in Section 3.3. As mentioned in that section, there

were two test routes in the vehicle test: one was an open sky route, the other was in

111

foliage. Only the section A of the open sky test (which contains mostly open sky data), as

well as the foliage route, was used for the TLA system research. In the following, the

term “open sky route” will only mean the section A of the open sky route.

4.4.2.1 Open Sky Test

For the open sky test, its test trajectory, reference velocity, and attitude are shown in

Figures 3.6 to 3.8, respectively. Its GPS satellite skyplot and signal C/N0 variations are

shown in Figure 4.8 and Figure 4.9, respectively. Figure 4.8 shows that in the open sky

test there are seven SVs in view with an elevation greater than 5 degrees. From Figure 4.9,

it can be seen that the carrier to noise power densities (C/N0) of the satellites in the open

sky test are high and vary by less than 5 dB-Hz for most satellites most of time. Only the

C/N0 of PRN 27 is low, but still more than 40 dB-Hz most of time. The C/N0 of PRN 11

has some large, but very short attenuations.

Figure 4.8 GPS Satellite Skyplot of Open Sky Test

112

Figure 4.9 Satellite C/N0 Variations of Open Sky Test

4.4.2.2 Foliage Test

In the foliage test, the test trajectory, reference velocity, and attitude are shown in Figure

4.10, Figure 4.11, and Figure 4.12, respectively. Figure 4.10 shows that the distance

traveled in the south-north direction was more than 2 km, and about 2 km in the east-west

direction. The vertical variation was more than 15 m. Figure 4.11 shows that the

maximum horizontal velocity was about 15 m/s, and the maximum vertical velocity about

0.8 m/s. From Figure 4.12, it can be seen that the pitch mostly ranged between -2 and 2

degrees, and the roll mostly between 0 and 4 degrees. The RMSs of the pitch and roll

were about 1.0 and 2.1 degrees, respectively.

113

Figure 4.10 Trajectory of Foliage Test

Figure 4.11 Reference Velocity of Foliage Test

114

Figure 4.12 Reference Pitch, Roll and Azimuth of Foliage Test

The GPS satellite skyplot and C/N0 variations of the foliage test are shown in Figure 4.13

and Figure 4.14, respectively. From Figure 4.13, it can be seen that there were nine SVs

in view with an elevation greater than five degrees. Figure 4.14 shows that the C/N0

variations of satellites are dramatic, and can reach as much as 20 dB-Hz.

115

Figure 4.13 GPS Satellite Skyplot of Foliage Test

Figure 4.14 Satellite C/N0 Variations of Foliage Test

4.5 Data Processing and Test Scenario

Data processing was performed with the TLA GPS/RIMU software developed from the

University of Calgary’s GSNRx™ software receiver (Petovello et al 2008b). In order to

116

implement TLA GPS/RIMU integration, the GSNRx™ software receiver was modified

by the author of this dissertation to include IMU data processing, and to implement the

TLA architecture, as shown in Figure 4.1. Based on the research of this dissertation, only

the receiver’s GPS L1 C/A code post-mission capabilities were used. For the translation

stage test data, a 2A1G (HG1700) configuration was used. Its mechanization equations

and corresponding error model were developed based on the stage test conditions, and

given in Petovello et al (2007).

For the vehicle test data, two RIMU configurations (i.e. 3A1G and 2A1G) were used for

each grade of IMU (tactical and MEMS). In addition to assessing the performance of the

TLA GPS/RIMU, the performance of TLA GPS/RIMU with ALFs was evaluated. For the

purpose of this work, the TLA GPS/RIMU software was further modified by the author

of this dissertation to include the adaptive loop filter architecture described in Section

4.3.2. The receiver’s output rate was set to 10 Hz and the IMU data was processed at 100

Hz. In order to facilitate comparison, the GPS measurement noise variance (used to

compute the navigation solution) was the same for all tests, i.e. 100 (m2) for pseudorange,

0.04 (m/s)2 for pseudorange rate.

A necessary part for implementing the ALF algorithm described in Section 4.3.2 is to

properly model the oscillator errors. To this end, the Allan parameters of the OCXO

oscillator used were chosen from Table B.1. The power curve of the random vibration

for calculating vibration-induced oscillator phase jitter was chosen from Table B.2 and

multiplied by 0.01. This scaling was necessary to yield a good approximation with the

117

real power spectral density (PSD) curve shown in Figure 4.15, which is obtained from the

vehicle test data described in Section 3.3. This is because the test in this dissertation is for

a land vehicle, but Table B.2 was obtained from a test for a turbojet transport aircraft

(Gebre-Egziabher et al 2005). As per Gebre-Egziabher et al (2005), the Allan parameters

of the satellite oscillator were chosen as those of a temperature compensated crystal

oscillator (TCXO) multiplied by 0.01. With reference to Figure 4.15, the “spline” stands

for the cubic spline fit interpolation method provided by Matlab.

Figure 4.15 Power Curve of Vehicle’s Random Vibration

4.6 Evaluation of TLA GPS/RIMU

The evaluation of the TLA GPS/RIMU is performed with both the translation stage test

data and the vehicle test data. Thus the advantages of the TLA system can be investigated

completely, i.e. in both ideal (or near ideal) and real situations. To identify the TLA

system’s advantages, the test results of the TLA system are compared with those of

corresponding tight system (i.e. no Doppler aiding for the GPS receiver). In order to

simplify analysis, only some representative test results are shown in the following.

118

4.6.1 Test Results with Translation Stage Test Data

With translation stage test data, the TLA GPS/RIMU (for the translation stage test, only

2A1G (HG1700) is used) is investigated from two aspects: PLL tracking performance

and system navigation performance. To investigate the PLL tracking performance, PRN

19 is chosen as an example. The reason for choosing PRN 19 is that PRN 19 has the

largest or near largest sensitivity to the stage motion since it has a low elevation angle

(about 21 degrees), and is the closest satellite with an azimuth in the north-south direction

(its azimuth is about 317 degrees, see Figure 4.6). The tracking performances of the TLA

and tight system for PRN19 are shown in Figure 4.16. In this figure, the phase lock

indicator is a value which is equal to the normalized estimate of the cosine of twice the

carrier phase error and is used to show the phase lock situation (Van Dierendonck 1996).

The phase lock indicator falls in the range of ±1, with +1 indicating perfect phase

tracking. From Figure 4.16, it can be seen that even though the two systems of TLA and

tight have the same PLL noise bandwidth (4 Hz), compared to the tight system, the TLA

system can improve the tracking performance. That is, the phase lock indicator (PLI) has

a value nearer to one, and the Doppler shift has a smaller error. From Figure 4.16, it also

can be seen that the C/N0 is attenuated at about GPS time 505960 s. Actually the GPS

signal is attenuated at GPS time 505956 s, and the attenuation is increased at a rate of

1 dB/s to a maximum of 40 dB. The C/N0 curve shows that the TLA system can give a

valid C/N0 estimate for longer time (i.e. for the signal with lower C/N0). This means that

the TLA system can track the signal with lower C/N0.

119

The forward velocity errors of the TLA and tight system are shown in Figure 4.17. From

this figure, it can be seen that the TLA system has much smaller velocity error,

specifically for TLA system, the RMS forward velocity error is 0.02 m/s; for tight system,

it is 0.20 m/s. More test results and analyses for the translation stage test were given in

Petovello et al (2007), and some conclusions were obtained as follows: the velocity

solution obtained from the TLA GPS/2A1G (HG1700) is about 60-90% more accurate

than the corresponding tight system solution. Furthermore, the TLA GPS/2A1G

(HG1700) is found to provide roughly 5 dB of sensitivity improvement relative to tight

GPS/2A1G (HG1700).

Figure 4.16 Tracking Performance Comparison of TLA and Tight GPS/2A1G

(HG1700) for PRN19

120

Figure 4.17 Forward Velocity Errors of TLA and Tight GPS/2A1G (HG1700)

4.6.2 Test Results with Vehicle Test Data

With vehicle test data, the TLA GPS/RIMU is evaluated both in terms of the number of

SVs used in the navigation solution and the system navigation performance. The number

of SVs used in the navigation solution also can be used as a criterion to assess the

tracking performance, i.e. generally, the more SV’s used in the navigation solution, the

better the tracking performance. In order to simplify analysis, only the test results of

GPS/3A1G (Crista) are shown in the following. To investigate the advantages of the TLA

system, the test results of the TLA system and the corresponding tight system are

displayed together. For the TLA GPS/3A1G, a noise bandwidth of 3 Hz was used, and a

second-order loop filter was used for the reasons discussed in Section 4.3.2. For the tight

GPS/3A1G, the noise bandwidth was set to 8 Hz and a third-order loop was used.

With the open sky data, the number of SVs used in the navigation solution of GPS/3A1G

(Crista) is shown in Figure 4.18. The same plot for the foliage data is shown in Figure

121

4.19. Both of the two figures show that the TLA system has more SVs used in the

navigation solution, especially for the foliage case. This means that with Doppler aiding,

the tracking performance of the GPS receiver is improved, and the improvement is more

obvious in the foliage case. This is because the PLL phase error is more sensitive to the

noise bandwidth when the C/N0 is low. That is, when the C/N0 is low, narrowing the

noise bandwidth can significantly reduce the PLL phase error (Ward et al 2006). This

result is consistent with the theory.

The navigation performance of the GPS/3A1G (Crista) is summarized in Table 4.1 for the

open sky case, and Table 4.2 for the foliage case. From the two tables, it can be seen that

the TLA GPS/3A1G has a better performance than the corresponding tight system in

terms of velocity and attitude, especially for velocity. Compared to the tight system, the

RMS three-dimensional (3D) velocity error of the TLA system is reduced by 54% for the

foliage case; for the open sky case, it is only a bit. In the both cases, both pitch and roll

error are reduced a little, if at all, but the azimuth error is noticeably reduced. Specifically

in foliage case, the RMS azimuth error of TLA GPS/3A1G (Crista) is reduced by 26%

compared to the tight GPS/3A1G (Crista). Furthermore, the attitude error reduction of the

TLA system is more obvious in the foliage case than in the open sky case. Also, it can be

seen that compared to the foliage case, the open sky test has much smaller velocity error

but has a greater pitch error. This is because the two test routes have different local

terrain. The position errors of the TLA GPS/3A1G with open sky data and foliage data

are summarized in Table 4.3. This table shows that the position errors are only few

metres (less than 10 m in 3D) for the both test routes. These values are only used to show

122

how the magnitude of position errors vary with the two test routes (i.e. the open sky and

foliage). These position errors mainly come from the GPS solution. So if the position

error needs to be reduced, a DGPS and/or a real time kinematic (RTK) technique needs to

be applied.


GPS/3A1G (Crista) with Open Sky Data


GPS/3A1G (Crista) with Foliage Data

123


Open Sky Data

RMS Attitude Error (Deg) RMS Velocity Error (m/s) Integration

Strategy Pitch Roll Azimuth East North Up

TLA GPS/3A1G 0.65 0.58 2.54 0.04 0.04 0.03

Tight GPS/3A1G 0.65 0.59 2.96 0.04 0.05 0.04


Foliage Data



TLA GPS/3A1G 0.33 0.63 2.71 0.07 0.08 0.03

Tight GPS/3A1G 0.46 0.76 3.64 0.11 0.18 0.11

Table 4.3 Position Error of TLA GPS/3A1G (Crista)

RMS Position Error (m) TLA GPS/3A1G

East North Up

With Open Sky Data 0.83 1.69 6.27

With Foliage Data 4.18 1.57 7.08

4.6.3 Summary

With both the translation stage test data and the vehicle test data, the performance of the

TLA GPS/RIMU was investigated in this section. In order to identify the TLA system’s

advantages, results of the TLA system were compared with those of the corresponding

tight system. Through a series of the performance assessments in terms of tracking ability

and navigation solution, the following conclusions can be drawn:

• In both ideal and real situations, i.e. on both the roof of a building and a real road,

the TLA GPS/RIMU performed better than the corresponding tight system in

terms of tracking ability and navigation solution.

124

• With Doppler aiding, the tracking performance of the GPS receiver was improved.

Specifically, there were more satellites used in the navigation solution, and the

Doppler shift measurement error was reduced.

• With Doppler aiding, the navigation performance of the TLA GPS/RIMU was

improved, i.e. the 3D RMS velocity error was reduced by 54% (for the foliage

case); both the pitch and roll error were reduced a little, if at all, but the azimuth

error was more significantly reduced.

4.7 Evaluation of TLA GPS/RIMU with ALFs

Following the evaluation of TLA GPS/RIMU, the evaluation for TLA GPS/RIMU with

ALFs is performed with vehicle test data. In the following, the term “TLA GPS/RIMU”

means the PLL tracking loops of the GPS receiver have a constant noise bandwidth (CNB)

loop filter, whereas the term “TLA GPS/RIMU with ALFs” means that the PLL tracking

loops have an adaptive loop filter, as described in Section 4.3.2. The objective of this

section is to investigate if the ALF algorithm is valid (i.e., can provide a proper noise

bandwidth) and if the performance of the TLA GPS/RIMU with ALFs can be further

improved, compared to with the constant noise bandwidth loop filter (CNBLF). For this

purpose, in the following tests, both open sky and foliage test data were used for the

evaluation of the TLA system with ALFs in terms of tracking and navigation

performance improvement. In order to facilitate comparison, for the TLA GPS/3A1G, the

(constant) noise bandwidth was chosen as 5 Hz for both grades of IMUs and both test

routes (open sky and foliage). This value was selected based on the largest bandwidth

computed using the ALF approach (about 5 Hz). In this way, the constant bandwidth

results are “optimal”, in the sense that the smallest bandwidth that captures the receiver

125

dynamics is used. Similarly, for the GPS/2A1G system, the CNB was set to 10 Hz based

on the maximum computed value of about 10 Hz when the ALF was used. In the

following, nB is used to denote the tracking loop noise bandwidth.

4.7.1 Test Results with Open Sky Test Data

Before investigating the tracking and navigation performance of the TLA system with

ALFs, variations in the attitude and velocity error of the TLA system are briefly reviewed.

4.7.1.1 Velocity and Attitude Error

For the TLA GPS/RIMU with ALFs, the velocity and attitude error of the 3A1G (Crista)

are shown in Figure 4.20, and the corresponding error for the 2A1G (Crista) are shown in

Figure 4.21. From these two figures, it can be seen that the TLA GPS/2A1G has more

vertical velocity error, which will be quantified later.


Open Sky Data

126


Open Sky Data

4.7.1.2 PLL Tracking Performance Comparison between ALF and CNBLF

In the following, PRN 27 is taken as an example to show the variations of phase lock

indicator (PLI), Doppler shift error, and noise bandwidth. The reason for choosing PRN

27 is that it has high sensitivity to the vehicle’s motion (most in the horizontal) since it

has low elevation angle (about 23-18 degrees), as shown in Figure 4.8. Therefore, with

PRN 27 it is easy to investigate not only the effect of the vehicle’s dynamics on the noise

bandwidth selection, but also the effect of the ALF on PLL tracking performance. The

PLI, Doppler shift error, and noise bandwidth of PRN 27 in the TLA system are shown in

Figure 4.22 and Figure 4.23. Figure 4.22 shows the tracking performance comparison of

the TLA GPS/3A1G (Crista) with and without ALFs (“without ALFs” means “with

CNBLFs”). Figure 4.23 shows the same plot for the GPS/2A1G (Crista) configuration.

127

From Figure 4.22, it can be seen that the Doppler error is reduced for the ALF case.

Specifically, the RMS Doppler error with and without ALF is 0.23 Hz and 0.28 Hz,

respectively. The phase lock indicator is not obviously improved. In fact, the PLL phase

error is not necessarily reduced with the ALF, i.e. the PLI value is not necessarily

increased (to approach 1) with the ALF if it is already greater than 0.9 (i.e. the PLL is

locked (Ward et al 2006; Van Dierendonck 1996)). This can be explained from the ALF

design strategy as given in Section 4.3.2. In the ALF design, when a signal is strong, the

lowest bandwidth for which the PLL is locked is selected to minimize the overall Doppler

error. As a result, the selected noise bandwidth (or the ALF) only promises to allow the

PLL locked, it does not promise to necessarily increase the PLI value (i.e. to reduce the

phase error). In addition, it is noted however that the phase lock in Figure 4.22 is

occasionally degraded (i.e. PLI is much less than 0.9) over short time periods. This may

be caused by unmodeled or unpredicted system errors, such as pitch and roll modeling

errors in the reduced IMU and GPS multipath error. Figure 4.22 also shows the calculated

noise bandwidth of the ALF for PRN 27, which is smaller than the corresponding

constant noise bandwidth, as expected. From Figure 4.23, just like the 3A1G, similar

conclusions can be obtained for the 2A1G configuration. Namely, the RMS Doppler

error is reduced from 0.54 Hz to 0.26 Hz and the phase lock is improved to some extent.

However, it is noted that the noise bandwidth in 2A1G is wider than in the 3A1G case.

This is expected because of the unmodeled vertical acceleration when only using two

accelerometers.

128

The RMS Doppler errors for all tracked satellites are summarized in Table 4.4 for the

3A1G, and Table 4.5 for the 2A1G. From these two tables, it can be seen clearly that with

ALF, the Doppler errors are reduced compared to with CNBLF, as expected. In Table 4.5,

for PRN 11, its RMS Doppler error with CNBLF (Bn=10 Hz) is much bigger than others,

i.e. the RMS Doppler error is 1.20 Hz for PRN 11, but the maximum Doppler error is

0.54 for others. This is caused by a few seconds of sudden signal attenuation, resulting in

an about 10 Hz frequency jump in the neighbourhood of GPS time 242300 s. This sudden

signal attenuation in PRN11 also caused a Z-count error in navigation data decoding in a

stand-alone GPS receiver test.


ALFs for Open Sky Test and PRN 27

129


ALFs for Open Sky Test and PRN 27

Table 4.4 RMS Doppler Errors of Tracked SVs in TLA GPS/3A1G (Crista) with

and without ALFs for Open Sky Test

RMS Doppler Error per PRN (Hz) Tracking Loop

Configuration 8 11 17 26 27 28 29

ALF 0.15 0.19 0.14 0.17 0.23 0.14 0.16

CNBLF (Bn=5Hz) 0.17 0.23 0.15 0.19 0.28 0.16 0.19


and without ALFs for Open Sky Test


Configuration 8 11 17 26 27 28 29

ALF 0.18 0.23 0.16 0.19 0.26 0.16 0.19

CNBLF (Bn=10Hz) 0.28 1.20 0.22 0.32 0.54 0.21 0.33

4.7.1.3 Navigation Performance Comparison between ALF and CNBLF

The statistical results of the TLA GPS/RIMU with ALFs are summarized in Table 4.6

and Table 4.7. Table 4.6 is the velocity and attitude error of TLA GPS/RIMU (Crista)

with and without ALFs. Table 4.7 shows the same value for the HG1700 IMU. From

Table 4.6, it can be seen that for the MEMS IMU, TLA GPS/3A1G with ALFs has

130

similar navigation performance as with CNBLFs ( 5=nB Hz). Only the north velocity

error of the ALF case is a little bit smaller than that of the CNBLF case, but the azimuth

error is a little bit larger This is very likely because in the open sky test, the GPS signals

are strong as shown in Figure 4.9, so a small reduction in noise bandwidth cannot

significantly improve the navigation performance. Similar conclusions can be drawn for

the tactical-grade inertial unit (see Table 4.7). The fact that the azimuth error of the

3A1G (Crista) with the ALF is a little bit bigger than that with the CNBLF is possibly

caused by the correlated Doppler shift measurement error or occasional factors, such as

the vertical gyro drift estimation situation before a stop. This needs to be further

investigated with more test data in the future. The reason for choosing the correlated

Doppler shift measurement error as a possible source causing a larger azimuth error is

explained as follows: the Doppler shift measurement error can be correlated in time when

the noise bandwidth of the PLL is narrowed. And the more the noise bandwidth is

narrowed, the stronger the measurement error is correlated (Gelb 1974). This can affect

the integration Kalman filter’s stability or the integration system’s performance (Bullock

et al 2006).

For the 2A1G configuration, navigation performance can be improved by using the ALF

approach compared to the CNBLFs ( 10=nB Hz) approach for both the MEMS and

tactical IMUs. For the MEMS IMU, the 3D RMS velocity error of GPS/2A1G with ALFs

is reduced by 13%. The pitch and roll errors almost keep the same values, only the

131

azimuth error is reduced somewhat. For the tactical-grade IMU, similar conclusions can

be obtained, with a 3D velocity error reduction of 15%.

Comparing the 3A1G and 2A1G results, it can be seen that the 3A1G case has smaller

velocity errors and similar attitude error compared to the 2A1G case. This applies to both

grades of IMUs and for both ALF and CNBLF. The improvement with the 3A1G case

arises because, no matter whether ALFs are used or not, the system has smaller noise

bandwidths because the extra accelerometer is better able to compensate for the receiver

dynamics. In turn, the navigation solution performance is improved. It is noted that the

improvement with the 3A1G case in the TLA GPS/RIMU is different from that in the

loose GPS/RIMU (see Chapter Three). In the TLA GPS/RIMU, the improvement with

the 3A1G case comes from not only the vertical accelerometer but also the narrower

noise bandwidth of the PLL. As a result, in the TLA GPS/RIMU, not only the vertical

velocity error but also the horizontal velocity error is reduced. However, in loose

GPS/RIMU (see Tables 3.1 and 3.2), only the vertical velocity error is reduced with

3A1G case. In this light, for TLA GPS/RIMU, the 3A1G configuration is a better choice

since it can allow a narrower noise bandwidth, resulting in not only a smaller vertical

velocity error but also a smaller horizontal velocity error.

132


ALFs for Open Sky Test

RMS Attitude Error

(Deg)

RMS Velocity Error

(m/s) Integration


TLA GPS/3A1G with ALFs 0.65 0.57 2.59 0.04 0.04 0.03

TLA GPS/3A1G (Bn=5Hz) 0.66 0.57 2.25 0.04 0.05 0.03


TLA GPS/2A1G (Bn=10Hz) 0.66 0.59 2.66 0.05 0.05 0.05


without ALFs for Open Sky Test

RMS Attitude Error

(Deg)

RMS Velocity Error

(m/s) Integration



TLA GPS/3A1G (Bn=5Hz) 0.43 0.40 0.42 0.04 0.04 0.03


TLA GPS/2A1G (Bn=10Hz) 0.44 0.42 0.53 0.04 0.05 0.05

4.7.2 Test Results with Foliage Test Data

With foliage test data, TLA GPS/RIMU with ALFs has similar velocity and attitude error

variations as with open sky test data. As such, focus is given only to the tracking and

navigation performance analysis as in the open sky test.

4.7.2.1 PLL Tracking Performance Comparison between ALF and CNBLF

Figure 4.24 compares the tracking performance of TLA GPS/3A1G (Crista) with and

without ALFs for PRN 25. Figure 4.25 shows the same plot for the 2A1G (Crista) case.

In this test, PRN 25 is selected for the same reason as in the open sky test, i.e. PRN 25

has low elevation angle (about 30-22 degrees), as shown in Figure 4.13. Its C/N0 varies

dramatically, and it still can keep tracking the signal, as shown in Figure 4.14.

133

From Figure 4.24 and Figure 4.25, it can be seen that for both 3A1G and 2A1G with

ALFs, the Doppler errors are reduced. The improvement of the phase lock indicator is not

obvious for 3A1G, but obvious for 2A1G, just like for the open sky test. In this foliage

case, the test results of phase lock indicator are explained as follows: in the ALF design,

when a signal is weak, the bandwidth that minimizes the total PLL phase error is selected

(see Section 4.3.2). As a result, the selected noise bandwidth promises to have a

minimum PLL phase error. However, it is noted that when the total phase error vs. noise

bandwidth curve is relatively flat in a certain range of noise bandwidth, a small reduction

in noise bandwidth cannot decrease the total phase error significantly. In this light, for

3A1G case, since the noise bandwidth reduction in the ALF is small (only 1-2 Hz), the

improvement of phase lock indicator is not obvious. The RMS Doppler errors of all

tracked satellites are summarized in Table 4.8 for the 3A1G (Crista), and Table 4.9 for

the 2A1G (Crista). From these two tables, similar conclusions can be drawn as for the

open sky test.

134


ALFs for Foliage Test and PRN 25


ALFs for Foliage Test and PRN 25


and without ALFs for Foliage Test


Configuration 8 11 17 19 25 26 27 28 29

ALF 0.14 0.56 0.60 0.67 0.41 0.27 0.20 0.15 0.23

CNBLF (Bn=5Hz) 0.21 0.67 0.71 0.81 0.53 0.38 0.30 0.23 0.33

135


and without ALFs for Foliage Test


Configuration 8 11 17 19 25 26 27 28 29

ALF 0.26 0.60 0.70 0.80 0.49 0.38 0.33 0.27 0.35

CNBLF (Bn=10Hz) 0.44 1.47 1.46 1.60 1.17 0.83 0.64 0.48 0.71

4.7.2.2 Navigation Performance Comparison between ALF and CNBLF

The test results are summarized in Table 4.10 and Table 4.11. Table 4.10 lists the

velocity and attitude errors of the MEMS IMU with and without ALFs. Table 4.11 shows

the velocity and attitude errors of the tactical-grade IMU with and without ALFs. From

Table 4.10, it can be seen that for the MEMS IMU, with ALFs, the 3D RMS velocity

error of the 3A1G is reduced by 18%, the pitch and roll error are marginally reduced, and

the azimuth error is reduced by 17%. Further, comparing the test results of TLA

GPS/3A1G (Crista) with ALFs and with CNBLFs ( nB =3 Hz), as given in Table 4.10 and

Table 4.2 respectively, it can be seen that the TLA system with ALFs has similar results

as for the system with a CNB of 3 Hz. This noise bandwidth value of 3 Hz is the best that

can be obtained for the 3A1G case (for the data collected). It is obtained through a series

of noise bandwidth selection trials. This further demonstrates the ALF algorithm is valid

and can provide a proper noise bandwidth for the Doppler-aided PLL. For the 2A1G case

with ALFs, the 3D RMS velocity error is reduced by 19%, the pitch and roll error are

reduced a little bit, and the azimuth error is reduced by 16%.

For the tactical-grade IMU, similar conclusions can be drawn except for the azimuth error.

Specifically, for the 3A1G, the 3D RMS velocity error is reduced by 16%; for the 2A1G,

the velocity error is reduced by 19%. In the both cases (3A1G and 2A1G), azimuth error

136

is small and keeps almost the same value (about 0.3 degrees). This results from the fact

that the TLA system with the tactical-grade IMU takes much more azimuth information

from the IMU (with more accurate azimuth calculation) than from the GPS.

Table 4.10 Velocity and Attitude Error of TLA GPS/RIMU (Crista) with and

without ALFs for Foliage Test

RMS Attitude Error

(Deg)

RMS Velocity Error

(m/s) Integration



TLA GPS/3A1G (Bn=5Hz) 0.35 0.67 3.28 0.08 0.10 0.04


TLA GPS/2A1G (Bn=10Hz) 0.46 0.71 3.93 0.09 0.12 0.08


without ALFs for Foliage Test

RMS Attitude Error

(Deg)

RMS Velocity Error

(m/s) Integration



TLA GPS/3A1G (Bn=5Hz) 0.31 0.61 0.27 0.08 0.10 0.03


TLA GPS/2A1G (Bn=10Hz) 0.37 0.66 0.32 0.09 0.12 0.08


From the above test results, it can be concluded that with ALFs, the PLL tracking

performance of the TLA GPS/RIMU can be improved in terms of reducing the Doppler

error and providing better phase tracking, especially for the 2A1G case. However, some

unmodeled system errors may cause phase lock degradation during short time periods.

Compared to the CNBLF case, the navigation performance of the TLA GPS/RIMU with

ALFs is also improved. Specifically, the 3D RMS velocity error can be reduced by up to

19%. Similarly, pitch and roll errors also can be reduced, but only slightly. It was also

137

observed that azimuth error can be reduced by 17% for the MEMS IMU case. Comparing

the 3A1G and 2A1G results, it can be concluded that the 3A1G case has smaller velocity

error and similar or a little bit smaller attitude error than the 2A1G case (for the foliage

test and MEMS IMU, it has smaller azimuth error). This is expected because the

GPS/2A1G configuration inherently needs wider noise bandwidths (relative to the 3A1G

case), which results in lower navigation performance. It follows therefore, that the 3A1G

configuration is preferable. These conclusions are supported by the test results given in

Sun et al (2010).

Compared to the open sky test results, the foliage test results show more clearly that the

performance of the TLA GPS/RIMU with ALFs were improved relative to the CNBLF

case. This implies that the ALF is more efficient in the foliage case (with serious GPS

signal attenuations). This can be explained from the fact that when C/N0 is high (>40 dB-

Hz), the thermal phase jitter is small even if the noise bandwidth is wide (Ward et al

2006). Since the above improvements result only from implementing an adaptable noise

bandwidth, we conclude that the proposed approach is both valid and effective.

Correspondingly, the error model in Equation (4.5) and its underlying assumptions are

considered sufficiently appropriate for the application at hand.

Finally, it is noted that the variability of the terrain, the operational environment (e.g.,

foliage, urban, etc.) and sensor quality will all affect performance of a given system and

care should therefore be exercised when extrapolating the above results to other

situations/conditions. That said, it is expected the proposed algorithm will still yield

138

benefits in all cases with appropriate filter tuning, but further testing is required to

quantity the level of such improvements.

4.8 Summary

In this chapter, first the TLA GPS/RIMU system was introduced and analyzed. Its PLL

phase error was derived. Based on the theoretical analysis, an adaptive PLL loop filter

whose noise bandwidth can be adjusted according to the performance of the integration

system and GPS signal C/N0 was designed and applied to a TLA GPS/RIMU. Following

the theoretical analysis and the ALF design, first the evaluation for TLA system was

conducted with both the translation stage test data and the vehicle test data to investigate

the advantages of the TLA system. Then evaluation for TLA system with ALFs was

conducted to verify the ALF algorithm. In these evaluations, IMU configurations

consisted of either 3A1G or 2A1G and both tactical and MEMS grade IMUs were used.

From the evaluation results, some conclusions are drawn as follows:

• In both ideal and real situations, i.e. on both the roof of a building and a real road,

the TLA GPS/RIMU performed better than the corresponding tight system in

terms of tracking ability and navigation solution.

• With Doppler aiding, the tracking performance of the GPS receiver was improved,

with more satellites being tracked, and smaller Doppler shift measurement error.

• With Doppler aiding, the navigation performance of the TLA GPS/RIMU was

improved. The 3D RMS velocity error was reduced by 54% (for the foliage case);

both the pitch and roll error were reduced a little, if at all, but the azimuth error

was reduced obviously. Specifically in foliage case, the RMS azimuth error of

139

TLA GPS/3A1G (Crista) was reduced by 26% compared to the tight GPS/3A1G

(Crista).

• The ALF was valid and provided a proper noise bandwidth for the Doppler-aided

PLL of the TLA GPS/RIMU.

• With ALFs, the PLL tracking performance of the TLA GPS/RIMU was improved

in terms of reducing the Doppler measurement error and providing better phase

tracking, but some unmodeled system errors might cause phase lock degradation.

• With ALFs, the navigation performance of the TLA GPS/RIMU was improved,

especially for velocity. The 3D RMS velocity error was reduced by up to 19%.

The pitch and roll error was also reduced, but only slightly. The azimuth error was

reduced by 17% for MEMS IMU and the foliage test data.

• In TLA GPS/RIMU, the 3A1G case outperformed the 2A1G case since it allowed

a narrower noise bandwidth, resulting in not only a smaller vertical velocity error

but also a smaller horizontal velocity error.

140

Chapter Five: Ultra-Tight GPS/Reduced IMU

As discussed in Chapters One and Two, in a UT GPS/IMU integration, the tracking loops

of the GPS receiver are controlled by the navigation solution of the system thus resulting

in vector-based tracking loops (i.e. VBTLs), namely a VDLL for code correlation and

VFLL for carrier Doppler removal (Lashley et al 2008b; Petovello et al 2008a; Spilker

Jr.1996). Since the VBTLs are closed via the navigation solution of the UT system, it

offers some potential advantages over scalar-based tracking loops (i.e. SBTLs), such as

improved weak signal tracking, enhanced anti-jamming ability, and more rapid signal

recovery after a satellite blockage (Kennedy & Rossi 2008; Lashley & Bevly 2008a; Im

et al 2007; Petovello & Lachapelle 2006a). As a result, the UT system outperforms GPS-

only receivers (with SBTLs) in terms of tracking ability and navigation accuracy

(Kennedy & Rossi 2008; Lashley & Bevly 2008a). The ultra-tight architectures

mentioned in Section 1.2.4, namely VDF (Vector DLL and FLL) and VDCaP (Vector

DLL with cascaded PLL), also apply to reduced IMU systems. As with a full IMU, UT

GPS/RIMU has the same advantages, but the extent of the performance improvement is

expected to be lower because the user dynamics cannot be removed as much as for the

full IMU case.

With this in mind, the objectives of this chapter are threefold: 1) investigate the

performance of UT GPS/RIMU systems with VDF and with VDCaP, 2) present two

innovative algorithms/configurations for UT GPS/RIMU to improve system performance,

and 3) compare the UT system with the corresponding TLA system to identify their

respective advantages and disadvantages. Based on the above objectives, this chapter is

141

arranged as follows. First the UT GPS/RIMU is introduced, including the architectures

and the implementations of the UT system. Specifically, the UT GPS/RIMU with VDF

and UT GPS/RIMU with VDCaP systems are discussed. Second, an innovative

algorithm/configuration (i.e. a cascaded PLL and FLL, or CaPF, composite loop) is

developed to overcome the drawback of the cascaded PLL, thus producing more reliable

Doppler measurements. Further, another innovative algorithm/configuration –

reconfigurable tracking loops (RTLs) – for UT GPS/RIMU is developed to improve the

system performance in the case when a partial GPS outage occurs. Third, the UT system

is compared with the corresponding TLA system to identify their differences in both

tracking and navigation performance. Finally, following the above theoretical

development, evaluations for UT GPS/RIMU with VDF and UT GPS/RIMU with

VDCaP are performed with real data. Further, the test results of UT system and the

corresponding TLA system are compared to quantify their differences in tracking and

navigation performance.

5.1 UT GPS/RIMU

Any GPS receiver must have both a code tracking loop and a carrier tracking loop (Borre

et al 2007; Ward et al 2006). In a UT GPS/RIMU, these two tracking loops are

implemented with VBTLs, which are closed via the navigation solution of the UT system.

Thus the tracking loops of each channel are not independent, that is, the loops of different

channels are coupled through the navigation solution, or navigation loop. The VBTLs can

be implemented using either a traditional transfer function method (Kiesel et al 2007;

Pany et al 2005) or a state-space method (Petovello et al 2008a; Crane 2007; Ohlmeyer

2006). Furthermore, with the state-space approach, a Kalman filter can be designed and

142

used for the VBTLs (Petovello et al 2008a; Crane 2007). In this chapter, only the transfer

function approach is used. However, it is noted that if all noise sources are properly

accounted for, both methods yield the same result. The benefit of the transfer function

approach is that it does not require explicit consideration of intermediate states and thus

can lead to a simpler analysis. In the UT GPS/RIMU, code tracking can be implemented

with a VDLL and carrier tracking can be implemented with either a VFLL or a VPLL.

Since a VPLL is quite difficult to implement, as discussed in Chapter One, an

approximate VPLL, i.e. a cascaded PLL, is considered. Consequently, two kinds of UT

systems are implemented: one is UT GPS/RIMU with VDF (or simply called VDF); the

other is UT GPS/RIMU with VDCaP (or simply called VDCaP). These two UT systems

are discussed below.

5.1.1 UT GPS/RIMU with VDF

The motivation for researching the VDF is that the VDF is one kind of UT system, which

can be used in land vehicle navigation. Accordingly, the VDF research can provide some

useful results for practical land vehicle navigation system design. The VDF system has

been investigated in many publications including Lashley & Bevly (2008a), Soloviev et

al (2007), and Pany & Eissfeller (2006). It is a common configuration for implementing a

UT system. In order to understand and design the VDF system, the implementation and

the measurement thermal noise of the VDF are discussed as follows.

5.1.1.1 VDF Implementation

In a UT GPS/RIMU with VDF, the code loop is implemented with a VDLL and the

carrier loop is implemented with a VFLL. Its block diagram is shown in Figure 5.1. From

this figure, it can be seen that both the code NCO and the carrier NCO are controlled by

143

the output of the integration system (i.e. the output of the integration Kalman filter). The

pseudorange prediction error and carrier Doppler prediction error directly come from the

corresponding discriminators.

Figure 5.1 Vector-Based Tracking Loops of GPS Receiver (One Channel)

The block diagram of a VDLL is shown in Figure 5.2. With reference to this figure, 1z−

represents a unit delay in z-transform. From Figure 5.2, it can be seen that in the VDLL,

each channel’s NCO is only controlled by the code phase and code Doppler, both

calculated from the UT system and not the discriminator output. So the individual

(channel) tracking loop is open. The code phase and code Doppler used for controlling

the code NCO can be calculated with Equations (2.18) and (2.21), respectively. In the

calculation, the noise and the unknown (or unmodeled) terms are ignored. Since the

Doppler shift calculated from Equation (2.21) is a carrier Doppler, it needs to be

transferred to a code Doppler by dividing 1540 (for C/A code and carrier 1L , or, more

144

generally, the ratio of the carrier frequency to the code chipping rate). In the code NCO,

the calculated code phase is used to reset the output of the NCO; the calculated code

Doppler is used to control the NCO. The output of the code tracking loop is

ˆm m NCO Discρ τ ρ ρ= ⋅ = + (5.1)

where mρ is pseudorange measurement of the code loop, NCOρ is the pseudorange output

of the NCO, and Disρ is the smoothed pseudorange error output of the discriminator,

where cρ τ= ⋅ , τ is the corresponding code phase (expressed in time), c is the speed of

light. In practice, it is generally assumed that NCO Estρ ρ≈ , where Estρ is the calculated

pseudorange.

Figure 5.2 Block Diagram of VDLL’s Lock Loop (One Channel)

The block diagram of the VFLL is shown in Figure 5.3. rϕ is input carrier phase, ϕ is

output carrier phase, daf is Doppler aiding, Disf is smoothed Doppler error output of the

frequency discriminator, and ˆdf is the output of the tracking loop. Figure 5.3 shows that

145

in the VFLL, each NCO is only controlled by carrier Doppler aiding calculated from the

UT system. So the individual (channel) carrier tracking loop is open, just like in the

VDLL. The Doppler aiding is the same as that calculated in the VDLL, but expressed as a

carrier Doppler (not code Doppler). The output of the frequency tracking loop is

ˆdm d da Disf f f f= = + (5.2)

where dmf is the carrier Doppler measurement of the frequency loop.

Figure 5.3 Block Diagram of VFLL Tracking Loop (One Channel)

From the block diagrams of VDLL and VFLL, it can be seen that each discriminator (for

both VDLL and VFLL) has a prefilter used for smoothing the discriminator’s output, thus

providing a smoothed estimation error measurement (for pseudorange and Doppler shift).

This smoothed error measurement can be used as the measurement to the Kalman filter of

the UT GPS/RIMU (see Figure 5.1) since the measurement of the Kalman filter equals

the actual measurement value minus the estimated value (see Equation (2.29)). Generally

the output rate of the discriminator is high (for example, if the predetection integration

146

time is 20PITT = ms, the output rate is 50 Hz), whereas the measurement update rate of

the integration Kalman filter is low (for example, the update rate is typically 10 Hz or

less). In order to filter the measurement noise of a discriminator, a prefilter is therefore

required (each channel has one individual prefilter for code and one individual prefilter

for carrier). The prefilter can be implemented with different algorithms, such as an

averaging algorithm (Bullock et al 2006), a linear model fitting algorithm (Pany et al

2005), and a Kalman filter algorithm (Lashley et al 2008b; Crane 2007; Petovello &

Lachapelle 2006a). To simplify computation, an averaging algorithm is used in this

chapter for both VDLL and VFLL.

For the VDLL, the prefilter algorithm is

1

1( 1) ( )

L

Dis Dis

i

k kL iL

ρ ρ=

+ = +∑ (5.3)

where ( 1)Dis kρ + is the output of the prefilter (i.e. the smoothed pseudorange error

output), ( )Dis kL iρ + is the output of the code discriminator (i.e. the input of the prefilter),

L is the number of discriminator outputs in a measurement update period of the

integration Kalman filter, k is the discrete time of the prefilter’s output, kL i+ is the

discrete time of the discriminator’s output. For the VFLL, the averaging algorithm is

1

1( 1) ( )

L

Dis Dis

i

f k f kL iL =

+ = +∑ (5.4)

where ( 1)Disf k + is the output of the prefilter, and ( )Disf kL i+ is the output of the

frequency discriminator (see Figure 5.3).

147

5.1.1.2 VDF Measurement Noise

The motivations for discussing the VDF measurement noise are twofold: 1) the

measurement noise analysis can be used in the performance discussion of the VDF; and 2)

the measurement noise analysis also can be used in the performance comparison of the

UT and TLA systems. To simplify analysis, only the thermal noises of the VBTLs (i.e.

VDLL and VFLL) and the corresponding SBTLs (i.e. DLL and FLL) are discussed and

compared in the following.

Since in VDLL and VFLL the individual tracking loops are open, the measurement noise

of the discriminators cannot be suppressed through a locally-closed loop (i.e. an

individual loop). Fortunately the prefilter is used to filter out some measurement noise of

the discriminator. For a code loop, if the variance of the measurement thermal noise of

the discriminator (i.e. in the discriminator output) is 2

Disρσ , the variance of the

pseudorange measurement thermal noise of the locally-closed loop (i.e. DLL) is (Misra &

Enge 2001; Van Dierendonck et al 1992)

2 22DLL n PIT DisB Tρ ρσ σ= (5.5)

where nB is the noise bandwidth of the locally-closed code loop (i.e. a scalar-based

tracking loop), and PITT is predetection integration time. The calculation equation of

DLLρσ is given in Misra & Enge (2001). The variance of the pseudorange measurement

thermal noise of the VDLL (for each channel) can be obtained from the prefilter (with

averaging algorithm, see Equation (5.3)), and is given as

148

2 2 /VDLL Dis Lρ ρσ σ= (5.6)

This eqaution is obtained from the assumption that the output noises (at different times)

of the discriminator are independent (Misra & Enge 2001). Suppose 20PITT = ms,

0.05nB = Hz (for Doppler-aided code loop), and the measurement update rate of the

integration Kalman filter is 10 Hz, then 5L = (i.e. Kalman filter update period (100 ms)

divided by PITT (20 ms)), and 32 2 10 1 / 0.2n PITB T L−= × =≪ (where 2 n PITB T is the

coefficient of 2

Disρσ in Equation (5.5), and 1 / L is the coefficient of 2

Disρσ in Equation

(5.6)). So compared to a scalar-based code loop (i.e., a DLL), a VDLL has much greater

measurement thermal noise.

In a similar way, if the variance of the measurement thermal noise of a frequency

discriminator (i.e. in the discriminator output) is 2

fDisσ , the variance of the Doppler

measurement thermal noise of the FLL is (Misra & Enge 2001)

2 22fdFLL n PIT fDisB Tσ σ= (5.7)

The calculation equation of fdFLLσ is given in Equation (5.16). The variance of the

measurement thermal noise of the VFLL (for each channel) is

2 2 /fdVFLL fDis Lσ σ= (5.8)

Again, for 20PITT = ms, 5nB = Hz (for a Doppler-aided FLL), and 5L = ,

2 0.2 1 /n PITB T L= = is obtained. For these parameters, the VFLL and the FLL have the

same measurement noise variance.

149

5.1.2 UT GPS/RIMU with VDCaP

The motivation for the VDCaP research is the same as that of the VDF research, i.e. the

VDCaP is another kind of UT system, which can be used in land vehicle navigation, and

the VDCaP research can provide useful information for practical land vehicle navigation

system design. Furthermore, for carrier phase loop, a cascaded PLL (CaPLL) outperforms

a Doppler-aided PLL (DaPLL) in some cases such as weak GPS signal case, which will

be discussed below.

There are quite a few publications discussing the VDCaP approach, such as Petovello et

al (2008a), Crane (2007), and Ohlmeyer (2006). Since a PLL outperforms a VFLL or a

FLL in terms of Doppler estimation and navigation data bit estimation, i.e. the VFLL or

FLL can decrease velocity estimation quality, and increase the complexity in estimating

navigation bits (Kiesel et al 2007), the VDCaP is more popular than the VDF in

implementing a UT system. It is noted that the conclusion that a VFLL or FLL can

decrease velocity estimation quality compared to a PLL should be obtained from the

thermal noise and steady state of the tracking loop. If the Doppler jitter caused by both

the transient state of the tracking loop and the C/N0 variations is considered, this

conclusion should be amended. In the following, first the implementation of the VDCaP

is discussed including the cascaded PLL design. Then the Doppler measurement thermal

noise of the VDCaP, which is caused by the discriminator output noise, is derived and

compared with that of a VFLL.

150

5.1.2.1 VDCaP Implementation

The VDCaP has a similar block diagram as that in Figure 5.1. In a UT GPS/RIMU with

VDCaP, the code loop is implemented with a VDLL and the carrier loop is implemented

with a cascaded PLL. As discussed in some papers such as Petovello et al (2008a), Crane

(2007), and Ohlmeyer (2006), the cascaded PLL can be implemented with a local

Kalman filter and the carrier NCO is controlled by both the local filter and the navigation

filter. Furthermore, the local filter is reinitialized after each navigation filter cycle, i.e. the

state variables of the local filter are set to zero after each measurement update of the

navigation filter (Crane 2007). In this section, the VDLL of the VDCaP is the same as

that of the VDF. The CaPLL is implemented with a traditional tracking loop and is

similar to the Doppler-aided PLL discussed in Chapter Four. It differs from the DaPLL

because the CaPLL is reset under some conditions, as described in Crane (2007). The

block diagram of the CaPLL is shown in Figure 5.4. From this figure, it can be seen that

the Doppler measurement of the CaPLL is

ˆ( ) ( ) ( ) ( )dm d da LFf k f k f k f k= = + (5.9)

where LFf is the Doppler error output of the loop filter, k is discrete time. Other variables

are defined in Equation (5.2). When the loop filter is reset, the state variables of the filter

are set to zero.

151

Figure 5.4 Block Diagram of Cascaded PLL (One Channel)

For a second-order loop filter (used in the carrier phase loop in this Chapter), as shown in

Figure 5.5, resetting state variables to zero means that ( ) 0LFf k = and ( ) 0sumf k = (see

Crane (2007)). With reference to Figure 5.5, ( )sumf k is the output of the integrator of the

loop filter, and ( ) ( ) ( 1)re k k kϕ ϕ ϕ= − − is the error signal of the system (Misra & Enge

2001), where ( )r kϕ is the input phase of the phase tracking loop, ( 1)kϕ − (for initial state

ˆ( 1) (0)rϕ ϕ− = ) is the output phase of the loop. ξ is the damping ratio of the second-order

system, nω is the undamped natural frequency. The output Doppler of the second-order

loop filter, ( )LFf k , can be written as

2( ) ( ) 2 ( ) ( 1) ( ) 2 ( )LF sum n sum PIT n nf k f k e k f k T e k e kϕ ϕ ϕξω ω ξω= + = − + + . (5.10)

Before proceeding, it is worth noting that the loop filter is effectively tracking the error in

the aided Doppler data. As such, under nominal conditions when the INS is well

calibrated, ( )LFf k will be close to zero (if Doppler aiding has no error). More generally,

152

( )LFf k will consist of two parts: one is the Doppler aiding error and the other is an

“extra” Doppler which is generated to drive the difference between in the local and

incoming phase to zero according to the designed control law of the loop (Nise 2000;

Ogata 1997). Generally the control law controls the difference between in the local and

incoming phase to zero quickly. When the output phase completely follows the incoming

phase (i.e. the DaPLL is completely stabilized or in steady state), this extra Doppler value

will approach to zero (Ogata 1997). Thus the ( )LFf k will be equal or approximately

equal to the Doppler aiding error (with very small error).

Figure 5.5 Block Diagram of Second-Order Loop Filter

Furthermore, from Equation (5.10), it can be seen that the output of the loop filter is not

only related to the output of the integrator of the loop filter, but also related to the error

signal ( )e kϕ . In the DaPLL, ( )LFf k serves the purpose of both compensating the

Doppler aiding error (i.e. the error of daf ) and eliminating the error signal ( )e kϕ , as

discussed above. If the error signal ( )e kϕ is caused by the system’s dynamics, ( )LFf k

controls ( )kϕ to follow the ( 1)r kϕ + (see Figure 5.4). In this case, although ( )LFf k is not

completely equal to the Doppler aiding error (i.e. it contains an extra Doppler value for

153

controlling the output phase, as discussed above), the loop filter should not be reset. This

is because the two parts of ( )LFf k are both useful: one is for compensating the Doppler

aiding error, the other is for controlling the output phase to converge to the input phase

quickly.

On the other hand, if ( )e kϕ is mostly caused by noise (thermal or other noise), ( )LFf k

will contain another Doppler value, which is caused by the noise alone. If the GPS signal

is strong enough, this Doppler noise term can be ignored, like in the above discussion.

However, for weak signals, this term can become significant and must be considered.

Suppose the Doppler aiding error compensation part of the ( )LFf k is , ( )LF DAEf k , the extra

Doppler value for control is , ( )LF Conf k , and the Doppler noise part is , ( )LF Noif k . In some

foliage or weak GPS signal cases, frequently , ( )LF Noif k is much larger than , ( )LF DAEf k

and , ( )LF Conf k (especially for wide loop noise bandwidth case), resulting in the phase

loop losing lock on the signal. In this case, the ( )LFf k might seriously deviate from

the , ( )LF DAEf k . In these cases, resetting the loop filter can constrain the deviation of

the ( )LFf k from the , ( )LF DAEf k (i.e. constrain the term , ( )LF Noif k ), helping the loop to

recover quickly.

From the above analysis, it can be seen that the criteria for the loop filter reset should be

carefully designed. The loop filter can be reset after every measurement update of the

navigation filter, as discussed in Crane (2007). This can disturb the process of controlling

154

the output phase ( 1)kϕ − to converge to the input phase ( )r kϕ (see Figure 5.4), resulting

in a delay in phase lock. In contrast, in order to overcome this disadvantage, the loop

filter can be reset according to the following criteria:

• In a strong signal case, the loop filter is not reset.

• In a weak signal case, when PLIPLI T< and dm da fdf f T− > , reset the loop filter.

Here, PLIT is a PLI threshold, dmf and daf are defined in Equation (5.2), and fdT is

a Doppler error threshold.

In practice, PLIT and fdT should be chosen carefully, for example, 0.3PLIT = and

1 2fdT = − Hz (they are empirical values). When the loop filter of a CaPLL is not reset in

a test, the CaPLL becomes a DaPLL.

From the above loop filter reset criteria, it can be seen that only in weak signal cases

and/or when the loop has lost lock and the deviation of the output of the loop filter from

the Doppler aiding error is beyond a certain value should the loop filter be reset.

Adopting such an approach will limit the deviation of the output of the loop filter from

the Doppler aiding error and help the loop recover quickly from a loss of lock, as

discussed above. Consequently, the CaPLL should outperform the DaPLL when the loop

filter reset is applied. That is why in some TLA system tests in this dissertation (e.g.

foliage test, see Section 5.7.2), the loop filter is also reset according to the same reset

criteria as that of the CaPLL.

155

5.1.2.2 Doppler Measurement Noise

The motivation for discussing the Doppler measurement noise of the CaPLL is that the

measurement noise analysis can be used in the navigation performance discussion of the

VDCaP, and in the performance comparison of the VDF and VDCaP. To simplify

analysis, only the Doppler measurement thermal noise of the CaPLL is discussed and

compared with that of the VFLL. Actually, in the Doppler thermal noise analysis, a

CaPLL can be equivalent to a PLL (with lower noise bandwidth compared to a PLL

without Doppler aiding). From this light, the Doppler measurement thermal noise of a

PLL is discussed below.

Compared to an FLL, the Doppler measurement thermal noise of a PLL is smaller (Ward

et al 2006). However, a comparison between a PLL and a VFLL has not been performed.

Although Kiesel et al (2007) gave the conclusion that a VFLL can decrease velocity

estimation quality compared to a PLL, no formula or more information was given. In this

section, the Doppler measurement noise comparison of a PLL and a VFLL will be made

from theoretical analysis and equations. In order to compare the Doppler measurement

noises of a PLL and a VFLL, the Doppler measurement thermal noise variance of the

PLL needs to be derived first. To simplify the derivation, a first-order loop is used, as

shown in Figure 5.6 (Misra & Enge 2001), since it is not easy to derive the Doppler noise

variance directly from a second-order loop. Although using the Doppler noise variance of

a first-order loop to replace that of a second-order loop is not perfect in the thermal noise

analysis, it still has some instructive significance to the thermal noise analysis of the

156

second-order loop. In this light, the Doppler thermal noise variance derived from a first-

order loop will be used in this chapter.

Figure 5.6 Block Diagram of First-Order Phase Lock Loop

In Figure 5.6, ϕη is the discriminator output noise (i.e. thermal noise), nϕ is the loop

output noise, and fdn is the Doppler measurement noise. The variance of ϕη is expressed

as (Misra & Enge 2001)

0 0

1 1( ) 1

2 / 2 /PIT PIT

VarT C N T C N

ϕη

= +

(5.11)

where C/N0 is the carrier to noise power density. For the phase tracking loop, the

variances of the input noise ϕη and the output noise nϕ have the following relationship

(Misra & Enge 2001)

( ) 2 ( )n PITVar n B T Varϕ ϕη= (5.12)

Since ( 1)kϕη + and ( )n kϕ are independent, the variance of ( 1)fdn k + is obtained as

( ) ( )2 2( ) ( ) ( ) 2 ( ) ( )fd n n n PITVar n Var n Var B T Var Varϕ ϕ ϕ ϕω η ω η η= + = + (5.13)

For a first-order loop, 4n nBω = (Ward et al 2006). Substituting this equation and

Equation (5.11) into Equation (5.13) yields

157

2

0 0

16 1 1( ) 1

/ 2 / 2

nfd n

PIT PIT

BVar n B

C N T C N T

= + +

(rad/s)

2 (5.14)

Expressing ( )fdVar n in Hertz yields

0 0

( ) 2 1 1 11

2 / 2 / 2

fd nfdPLL n

PIT PIT

Var n BB

C N T C N Tσ

π π

= = + +

(Hz)

(5.15)

where fdPLLσ is the Doppler measurement thermal noise standard deviation obtained from

the first-order PLL.

The analysis now shifts to computing the standard deviation of the Doppler measurement

thermal noise for a VFLL. In order to derive the thermal noise standard deviation of the

VFLL, the standard deviation of the Doppler measurement thermal noise of an FLL is

needed first and can be expressed as (Ward et al 2006)

0 0

41 11

2 / /

f n

fdFLL

PIT PIT

B

T C N T C N

ζσ

π

= +

(Hz)

(5.16)

where fζ is a coefficient that is usually selected as 1fζ = at high 0/C N , and 2fζ =

near threshold (Raquet 2004). With Equations (5.7), (5.8) and (5.16), the standard

deviation of the Doppler measurement thermal noise of the VFLL is obtained as

0 0

21 11

2 / /

f

fdVFLL

PIT PIT PITT L T C N T C N

ζσ

π

= + ⋅

(Hz)

(5.17)

With Equations (5.15) and (5.17), the Doppler measurement thermal noise standard

deviations of the PLL and the VFLL are drawn in Figure 5.7 as a function of tracking

loop bandwidth. From this figure, it can be seen that with low C/N0, the VFLL has much

158

more Doppler noise compared to the high C/N0. With high C/N0, the Doppler noises of

both the PLL and VFLL are low. For the VFLL, its measurement noise variance is not

related to the noise bandwidth (because the VFLL is an open loop, there is no loop filter

in it). This is also evident from Equation (5.17), which is only a function of L (smoothing

parameter). But the measurement noise of the PLL increases with the noise bandwidth.

Thus if the noise bandwidth of the PLL is greater than a certain value (for a given C/N0),

the measurement noise of the PLL will be greater than that of the VFLL.

Figure 5.7 Doppler Measurement Noise Variances of PLL and VFLL

5.2 CaPF Composite Loop for UT GPS/RIMU

In order to design the CaPF, first the Doppler measurement error of the CaPLL is

analyzed. It is noted that in the following Doppler measurement error analysis, only the

DaPLL is considered because the Doppler measurement error analysis for both DaPLL

and CaPLL are the same. As discussed in Section 5.1.2.1, the loop filter output of a

DaPLL consists of two parts: one is the Doppler aiding error, the other is an extra

Doppler value, which is used to control the output phase of the tracking loop to converge

to the input phase. This extra Doppler value causes Doppler measurement errors in the

159

DaPLL and CaPLL cases. In general, the larger the phase errors in the tracking loop (i.e.

the input phase minus the output phase), the larger the extra Doppler values, and the

larger the Doppler measurement errors. This is a drawback of the DaPLL. Actually the

PLL (without Doppler aiding) also has this same drawback.

In order to overcome this drawback of the CaPLL (or just DaPLL), a composite loop,

which consists of a CaPLL and an FLL discriminator, is developed in the hope of

providing more reliable Doppler measurements in both the PLL locked and unlocked

cases, thus improving the performance of the UT GPS/RIMU. This composite loop is

termed as CaPF, and the UT system is herein called UT GPS/RIMU with VDCaPF, that

is, a VDLL plus a CaPF.

The block diagram of the CaPF is shown in Figure 5.8. With reference to this figure,

“Disc.” is the abbreviation of “discriminator”, dmf is the Doppler measurement output,

the “Filter Reset Control” is the same with that of the CaPLL, as shown in Figure 5.4, but

the filter reset criteria are different. Specifically, the loop is reset under the following

conditions:

• After every measurement update from the navigation filter, when 0.9PLIPLI T< =

• In weak signal environments with the 2A1G configuration and when

0.3PLIPLI T< = and dm da fdf f T− > ( 1 2fdT = − Hz)

The motivation of choosing such loop filter reset criteria is to simulate a VFLL

when 0.9PLIPLI T< = in order to obtain a better Doppler measurement. More specifically,

160

when the PLI is less than 0.9 it suggests the phase is not being tracked very well. If no

actions were taken, the tracking loops would adjust to try to drive the phase error to zero,

and in the process generate Doppler measurements that are not only related to range rate.

Therefore the Doppler measurements are not accurate. If the Doppler measurements are

used to update the navigation solution, the navigation performance will be degraded. To

avoid this, the loop filter in the CaPF approach is reset such that LFf becomes zero, and

correspondingly, the NCO is driven only by the aiding Doppler. If the NCO is only

driven by the aiding Doppler, the output of the frequency discriminator is the error in the

aiding Doppler value. As will be shown below, this discriminator output can be filtered

and added to the aiding Doppler value in order to generate a Doppler measurement that

much more closely related to the range rate, that is, they are more accurate. In the process,

the expectation would be that the navigation solution is more accurate and thus the aiding

Doppler would also become more accurate with the ultimate benefit of improving signal

tracking.

However, the loop filter in the CaPF approach is not always reset. It is only reset when

the above loop filter reset criteria are satisfied. For example after every measurement

update from the navigation filter and when 0.9PLI < . If 0.9PLI < , at every

measurement update time or point, the loop filter is reset. But between two measurement

update points, the loop filter is not reset in order to allow the tracking loop a great

possibility to be locked. Thus the tracking loop of the CaPF is a DaPLL during two

measurement update points, but at the measurement update points, the carrier loop is

actually a VFLL. In order to obtain the measurement of Doppler aiding error during two

161

measurement update points, the Doppler aiding error needs to be calculated and is

discussed in the following. It is noted that only when the carrier loop is a VFLL, the

output of the frequency discriminator is the error in the aiding Doppler value.

Figure 5.8 Block Diagram of CaPF (One Channel)

From Figure 5.8, it can be seen that the Doppler measurement comes from either the

CaPLL or the frequency discriminator, which is controlled by a switch. If the Doppler

measurement comes from the CaPLL (the output is switched to upper branch), it is

calculated with Equation (5.9); otherwise the Doppler measurement comes from the

frequency discriminator (i.e. the Doppler aiding plus the smoothed calculated Doppler

aiding error, and the output is switched to lower branch), and is calculated with

162

( )( )1

( 1) (( 1) 1)

1 ˆ ( ) ( 1) ( 1)

dm da

L

Dis d da

i

f k f k L

f kL i f kL i f kL iL =

+ = + −

+ + + + − − + −∑ (5.18)

where the term ( )( )ˆ( ) ( 1) ( 1)Dis d daf kL i f kL i f kL i+ + + − − + − is the calculated Doppler

aiding error. It is explained as follows.

With reference to Figure 5.8, suppose the Doppler output of the loop filter, LFf , can be

expressed as (as discussed in Section 5.1.2.1)

, ,LF LF DAE LF Conf f f= + (5.19)

where ,LF DAEf is the Doppler aiding error compensation part of the LFf , and ,LF Conf is the

extra Doppler value for controlling the phase. Herein the Doppler noise part is ignored. In

order to simplify the math, the discrete time is omitted from the expression of the

variables. From Equations (5.9) and (5.19), the Doppler measurement of the CaPLL is

obtained as

, ,ˆd da LF da LF DAE LF Conf f f f f f= + = + + (5.20)

Subtracting the Doppler aiding value, daf , from both sides of Equation (5.20) yields

, ,ˆd da LF DAE LF Conf f f f− = + (5.21)

Since the Doppler output of the frequency discriminator is the Doppler error, i.e. the

input Doppler (true Doppler) minus the output Doppler, then the Doppler output of the

frequency discriminator is

,Dis LF Conf f= − (5.22)

163

It is noted that the true Doppler is ,da LF DAEf f+ , and the output Doppler of the CaPLL is

, ,da LF DAE LF Conf f f+ + (see Equation (5.20)). So the Doppler error is ,LF Conf− . Substituting

Equations (5.21) and (5.22) into the term ˆ( )dis d daf f f+ − yields

,ˆ( )Dis d da LF DAEf f f f+ − = (5.23)

Thus the term ˆ( )Dis d daf f f+ − is the calculated Doppler aiding error.

From Equations (5.18), (5.20) and (5.23), it can be seen that the calculated Doppler

measurement with Equation (5.18) equals the Doppler aiding plus the smoothed

calculated Doppler aiding error, whereas the Doppler measurement of the CaPLL equals

the Doppler aiding plus the Doppler aiding error and the extra Doppler value for control.

If the extra Doppler value for control is sufficiently large due, for example, to weak

signals or transient state, the Doppler measurement of the CaPLL will have more error

than that of Equation (5.18). This often happens in the case where the phase loop has lost

lock on the signal. Consequently, the Doppler measurement from the calculation of

Equation (5.18) (or just simply called “from the FLL discriminator”) can improve the

measurement. Stated differently, if the loop has lost lock on the signal, the input to the

loop will be mostly noise in weak signal case. Correspondingly, the output of the

discriminator will be mostly noise. By filtering the discriminator outputs, the goal is to

remove this noise, and thus regain tracking capability.

In contrast, if the phase loop is locked, the extra Doppler value for control should

approach zero. Therefore, the Doppler measurement from both the CaPLL and Equation

164

(5.18) will have the same value. But the Doppler measurement from the CaPLL has

smaller thermal noise than that from Equation (5.18) when the noise bandwidth is

sufficiently narrow (see Figure 5.7). For this reason, when the phase loop is locked, the

Doppler measurement should come from the CaPLL.

In order to take advantage of the Doppler measurement of the frequency discriminator,

the Doppler measurement output is switched according to the following criterion: when

0.9PLIPLI T≥ = , dmf is switched to CaPLL; else dmf is switched to frequency

discriminator. This is because if 0.9PLI < , the CaPLL tracking loop is considered

unlocked (Ward et al 2006; Van Dierendonck 1996), and the loop filter is reset. If the

Doppler measurement comes from Equation (5.18), it will have less Doppler error (i.e. it

does not contain the extra Doppler value for control). Thus the carrier loop will simulate

a VFLL. It is noted that this CaPF configuration can be used for any UT system, not only

for the UT GPS/RIMU.

5.3 Reconfigurable Tracking Loops for UT GPS/RIMU

In a UT system, the NCOs of the VBTLs are controlled by the navigation solution of the

UT system. If the navigation solution is not accurate enough to maintain the signal lock,

the VBTLs will fail, i.e. all tracking loops will break down. This is the main drawback of

VBTLs (Pany & Eissfeller 2006; Pany et al 2005). In contrast, SBTLs are not affected by

the navigation solution accuracy of the system. For a SBTL, even if there is no navigation

solution obtained, the SBTL can still keep tracking when the GPS signal is strong enough.

To overcome the drawback of the VBTLs, the VBTLs should be switched to the

corresponding SBTLs when the navigation solution accuracy of the UT system is

165

sufficiently poor. But the question arises then as to what conditions or criteria should be

used to switch the VBTLs to SBTLs? Pany et al (2005) used time to control the switch

from VBTLs to SBTLs during a GPS outage. This is not accurate enough for a UT

GPS/RIMU because the navigation solution accuracy is not only related to the GPS (or

the GPS outage duration time), but also related to the RIMU. From this viewpoint, in this

section, the estimated navigation performance of the UT GPS/RIMU is used to control

the switch between VBTLs and SBTLs, specifically, the switch between VDLL and DLL,

and the switch between CaPF and PLL (no Doppler aiding). Thus a reconfigurable

tracking loop, which can switch between VDLL and DLL, and switch between CaPF and

PLL, is obtained.

5.3.1 Switch Strategy between VDLL and DLL

In a DLL, the rule-of-thumb tracking threshold is (Ward et al 2006)

2/33 DRetDLLDLL ≤+= σσ (5.24)

where tDLLσ is thermal noise code tracking jitter (1σ), eR is dynamic stress error in the

DLL tracking loop, and D is early-to-late correlator spacing. In a VDLL, since the

individual tracking loop is open and the NCO is controlled directly by the navigation

solution of the UT system as shown in Figure 5.2, then, with reference to Equation (5.24),

the tracking threshold of a VDLL can be written as

3 3 / 2VDLL Est Estc Dρ ρσ σ= + ≤ (5.25)

where VDLLσ is the total pseudorange estimation uncertainty caused by the navigation

solution of a UT GPS/RIMU, Estρσ is the standard deviation of the pseudorange

estimation noise, and Estcρ is an empirical constant to account for unmodeled

166

pseudorange errors such as multipath and atmosphere delays . More specifically, Estcρ is

the pseudorange error which is not accounted for in the term Estρσ . The units of all

variables in Equation (5.25) are metres.

Based on Equation (5.25), the switch strategy between VDLL and DLL is designed such

that if 3 / 2Est Estc Dρ ρσ + ≤ , the code tracking loop is switched to VDLL; else the

tracking loop is switched to DLL. This is because if 3 / 2Est Estc Dρ ρσ + > , the VDLL may

fail.

For a given navigation position error, its projection onto the line-of-sight (LOS) of each

satellite is different, resulting in different pseudorange estimation errors for different

satellites. For the sake of simplification and to reduce calculations in the receiver, the

projection calculation is omitted and the maximum pseudorange error (calculated directly

from the navigation position error) is used to control the switch between DLL and VDLL,

i.e. only one switch control threshold is applied to all satellites. In practice, Estcρ can be

chosen according to different test environments, for example, in suburban area, Estcρ can

be chosen as 15 metres. This value is used herein because for the vehicle test used in this

dissertation, the 3D position error of the GPS position is about 10 metres. It is noted that

this value is selected to be larger than its theoretical value in order to make sure the

VDLL safely (conservatively) switches to the DLL before it fails. In contrast, Estρσ can

be calculated from the predicted covariance matrix, ( )−P , of the integrated navigation

167

Kalman filter. Suppose ( ) ( ) ( )T

r Eδ δ δ − = − − P r r , where ( )δ −r is navigation position

prediction error. In this case, ( )rδ −P is the sub-matrix of ( )−P corresponding to the

position states (see Section 4.3.1 for detail), and can be expressed as

11 12 13

21 22 23

31 32 33

( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( )

r

p p p

p p p

p p p

δ

− − − − = − − −

− − −

P (5.26)

The maximum pseudorange estimation noise variance is given by the sum of the

variances as

2

11 22 33( ) ( ) ( )Est P P Pρσ = − + − + − (5.27)

5.3.2 Switch Strategy between CaPF and PLL

The composite loop (CaPF) is closed with carrier phase. Actually its closed part is a

CaPLL. This CaPLL is both aided and reset with the Doppler aiding from the navigation

solution (see Section 5.2). With this in mind, if the Doppler aiding is not accurate enough,

the CaPLL will fail. Actually, the loop filter reset of the CaPLL is mainly affected by the

Doppler aiding error. In Section 5.2, the loop filter reset of the CaPF is discussed only in

the normal GPS case, i.e. no GPS outage. If there is a long enough GPS outage in a test,

the loop filter reset and the Doppler aiding of the CaPF should stop, and the CaPF should

be switched to a pure PLL (i.e. no Doppler aiding).

Actually, the effectiveness of Doppler aiding is mainly affected by the dynamics of the

aiding error. In other words, it is the dynamics of the aiding error that causes the Doppler-

aided PLL to fail. These dynamics are mainly induced by the attitude error of the

168

GPS/RIMU and the specific force measured by the RIMU (generally related to the

vehicle’s dynamics), especially during GPS outages. To illustrate this latter point,

suppose the specific force measured by the RIMU is the true specific force, bf , the true

rotation matrix from b-frame to ℓ -frame is bRℓ , and the estimated rotation matrix is bR

ℓɶ .

In this case, the dynamics of the Doppler aiding error is given as

( )b b b

b b b bδ = − = −f R f R f R R fℓ ℓ ℓ ℓ ℓɶ ɶ (5.28)

From this equation, it can be seen that for a given bf , the more the attitude error ( i.e.

b b−R Rℓ ℓɶ ), the more the dynamics of the Doppler aiding error (i.e. δ f

ℓ ). For a given grade

of inertial sensors, generally a full IMU has a less attitude error than an RIMU. Thus

there are less user dynamics left in the Doppler aiding error with a full IMU than with an

RIMU. Furthermore, if the accelerometer measurement error, bδ f , is considered, the

dynamics of the Doppler aiding error should contain another main term, b

bδR fℓ .

Generally, a large δ fℓ causes a large velocity error (Doppler aiding error), especially

during GPS outages (Farrell & Barth 1999). The Doppler aiding error can be obtained

from the pseudorange rate error divided by the signal wavelength (see Equation (2.26)).

In this light, only the pseudorange rate error (corresponding to Doppler aiding error) is

used to control the switch between CaPF and PLL, thus simplifying the switching

criterion. Based on this consideration, the criterion for applying a CaPF is selected as

rCaPF rEst rTρ ρ ρσ σ γσ= ≤ (5.29)

169

where rCaPFρσ (or rEstρσ ) is the pseudorange rate uncertainty standard deviation caused by

the navigation solution error of a UT GPS/RIMU, rTρσ is chosen as the pseudorange rate

uncertainty standard deviation in the case where there is no GPS outage, and γ is a

coefficient. In practice, rTρσ is chosen according to the performance of the GPS/RIMU,

γ is chosen according to the parameters of the tracking loop, such as the loop filter order

and noise bandwidth. For example, for vehicle test and MEMS IMU, rTρσ can be chosen

as 0.16 m/s (3D) (this is an empirical value, which was obtained from the test results of

Chapter Four), and γ can be chosen as 3 (for a second-order loop filter with a noise

bandwidth of 3 Hz for 3A1G case and 8 Hz for 2A1G case). This means that if the

pseudorange rate uncertainty standard deviation is more than three times the normal (no

GPS outage) pseudorange rate uncertainty standard deviation, it is considered that the

CaPF fails.

Based on Equation (5.29), the switch strategy between CaPF and PLL is designed such

that if rCaPF rTρ ρσ γσ≤ , the phase tracking loop is switched to CaPF; else the tracking loop

is switched to PLL.

Similar to the pseudorange estimation uncertainty calculation discussed in Section 5.3.1,

the projection of the velocity uncertainty onto the line of sight for each satellite is omitted

when computing the pseudorange rate estimation uncertainty. Correspondingly, the

maximum pseudorange rate uncertainty calculated directly from the navigation velocity

error covariance matrix is used to control the switch for every satellite. Specifically,

170

rCaPFρσ can be calculated from ( ) ( ) ( )T

v Eδ δ δ − = − − P v v , where ( )δ −v is the

navigation velocity prediction error. ( )vδ −P can be extracted from ( )−P (see Section

4.3.1 for detail), and can be expressed as

44 45 46

54 55 56

64 65 66

( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( )

v

p p p

p p p

p p p

δ

− − − − = − − −

− − −

P (5.30)

Again, the maximum pseudorange rate uncertainty variance is given by the sum of the

variances:

2

44 55 66( ) ( ) ( )rPF P P Pρσ = − + − + − (5.31)

5.3.3 Implementation of Reconfigurable Tracking Loops (RTLs)

Based on the above switching strategies between VDLL and DLL, and between CaPF

and PLL, the reconfigurable tracking loops are implemented as shown in Figure 5.9. It is

noted that in order to enhance the switch reliability, the switch back from SBTLs to

VBTLs is delayed one second from when the switch criterion is satisfied. This can

prevent frequent switches when operating near the threshold. This means that once a

tracking loop has been switched to a SBTL from a VBTL, its switch back should be done

cautiously. Correspondingly, the RTLs are mainly used in GPS outage cases. When there

are fewer than four satellites in view, the navigation solution error of the GPS/RIMU

might be too large to maintain signal lock using a VBTL approach. In this case, the

tracking loops are switched to SBTLs to let the satellites in view keep tracking, thus

improving the performance of the GPS/RIMU. Since the satellites in view still keep

171

tracking, when the loops are switched back to VBTLs, the reacquisition for these

satellites is avoided.

Figure 5.9 Flowchart of Reconfigurable Tracking Loop Implementation

5.4 Theoretical Comparison of UT and TLA GPS/RIMU

From the discussions of UT and TLA GPS/RIMU (see Chapter Four for TLA system), it

can be seen that UT and TLA GPS/RIMU are mainly different in the configurations of

tracking loops. In the UT system, tracking loops are closed via the navigation solution of

the UT system, whereas in the TLA system, tracking loops are closed via their respective

outputs. Because of the difficulties in implementing a VPLL, in this chapter a cascaded

PLL is used to replace the VPLL in the UT system, resulting in the VDCaP configuration

(see Section 5.1.2). In the TLA system, the code loop is a Doppler-aided DLL (DaDLL),

172

and the carrier loop is a DaPLL. Correspondingly, in the VDCaP approach, the code loop

is a VDLL, and the carrier loop is a CaPLL. Owing to the fact that the UT (i.e. VDCaP)

and TLA system have a same kind of carrier loops (i.e. phase lock loop), the comparison

between UT and TLA systems will be made in the VDCaP and the TLA system (if UT

and TLA systems have different kind of carrier loops such as a VFLL for UT system and

a DaPLL for TLA system, their comparison is not straightforward, and herein it is not

considered).

5.4.1 Comparison in Carrier/Code Tracking Loops

For carrier tracking loop, the DaPLL and the CaPLL are both Doppler-aided PLLs. In the

CaPLL case, however, the loop filter is reset according to some criteria (see Section

5.1.2.1). If the Doppler aiding is accurate enough, and the GPS signal is weak, the CaPLL

should outperform the DaPLL, at least in theory (see Section 5.1.2.1). This is because the

CaPLL can bound the Doppler measurement error, resulting from the loop filter reset.

The bounding for the Doppler error can benefit the tracking loop’s recovery from

temporary loss of lock, but it may disturb the process of the carrier phase convergence,

resulting in delay in phase lock, as discussed in Section 5.1.2.1. In this light, the CaPLL

underperforms the DaPLL in the case where the GPS signal is strong and/or the Doppler

aiding is not accurate enough (if the loop filter is still reset). As mentioned before, if the

loop filter of the CaPLL is not reset in a test, the CaPLL becomes a DaPLL.

For the code tracking loop, the UT system uses a VDLL, whereas the TLA system uses a

DaDLL. Sometimes the DaDLL is just simply called as DLL because in this dissertation,

the DLL is always aided with the code Doppler. For the DaDLL, it is a closed loop, and

173

can be modeled as a control system (see Section 2.3.2). With code Doppler aiding, the

noise bandwidth of the DLL can be narrowed, as discussed in DaPLL (see Chapter Four

for detail). So the DaDLL of the TLA system is a feedback control system with low noise

bandwidth. In contrast, the VDLL of the UT system can be considered as an open loop

(i.e. in the VDLL, the individual tracking loop is open, and the code loop is only closed

via the navigation solution of the UT system). The output of the code tracking loop is not

fed back directly to control the loop, but is instead fed to the navigation solution.

Consequently, the stability and the output of the open loop (i.e. the VDLL) are not only

related to the input of the open loop, but also related to the navigation solution of the UT

system (i.e. the control of the open loop). With this in mind, the comparison between the

VDLL and the DaDLL is made in the following four aspects: 1) pseudorange

measurement thermal noise; 2) system stability; 3) multipath alleviating ability; 4) weak

signal tracking ability.

The comparisons in the above four aspects are as follows:

• Compared to the VDLL, the DaDLL has a much smaller pseudorange

measurement thermal noise since it is aided with the code Doppler, and has low

noise bandwidth, as discussed in Section 5.1.1.2. For the given parameters of

Section 5.1.1.2, the pseudorange measurement thermal noise variance of the

VDLL is one hundred times that of the DaDLL.

• Since the DaDLL is a closed loop, it has the ability to track the signals with

unmodelled or unknown parameters such as time delays in code signals (Ogata

1997). But for the VDLL, since it is an open loop, if the unknown time delay of

174

the tracked signal is more than half of the early-to-late correlator spacing, the

VDLL will fail (see Equation (5.25)). Similarly, if the 3D position error of the UT

system is more than half of the early-to-late correlator spacing, which usually

occurs during partial GPS outages (for a complete GPS outage, there are no

satellites in view; this is not considered herein), the VDLL also will fail. From this

light, the DaDLL is more stable.

• Since the VDLL is an open loop, it cannot track signals with unmodelled or

unknown time delays which is more than half of the early-to-late correlator

spacing. A multipath signal can introduce unknown time delays (Misra & Enge

2001). If the unknown time delay is more than half of the early-to-late correlator

spacing, the VDLL will not track it, thus removing it from the navigation solution.

On the contrary, the DLL is a closed loop, it can track signals with unknown time

delays, such as multipath signals. But the DLL can alleviate multipath signals with

a narrow correlator (Misra & Enge 2001).

• The VDLL and DaDLL both have a weak signal tracking ability (Lashley & Bevly

2008a; Soloviev et al 2007; Gebre-Egziabher et al 2005). In the VDLL, since its

NCO is controlled by the navigation solution of the UT system, it achieves this

weak signal tracking ability (Lashley & Bevly 2009b). In contrast, the DaDLL

achieves this ability through narrowing its noise bandwidth.

5.4.2 Comparison in System Performance

Based on the above carrier/code tracking loop comparison, the system performance of the

UT and TLA GPS/RIMU are compared in terms of tracking ability and navigation

performance. The characteristics of the carrier/code tracking loops determine the tracking

175

ability and the navigation performance of the integration system. If only the difference of

UT and TLA system in code loop is considered, the TLA system is more stable than the

UT system because the DaDLL is more stable than the VDLL.

In the case where no partial GPS outage happens, the UT and TLA systems are compared

as follows. If the GPS signal is strong, the UT and TLA systems are supposed to have

similar tracking ability and velocity error. This conclusion is drawn based on the three

following facts: first the CaPLL and the DaPLL have similar tracking ability and

measurement noise in strong GPS signal case; second it is assumed that in the strong GPS

signal case, the calculated pseudorange (for code NCO control) has a small error, thus the

VDLL and DaDLL have a similar tracking ability, and; the thermal noise is low in strong

GPS signal case (Ward et al 2006). If the GPS signal is weak, both the code and carrier

loop of the UT and TLA system will be affected. Comparing the two systems’

performance in theory is not straightforward in this case. Accordingly, the performance

comparison of the two systems will be performed empirically with test results.

The comparisons for the tracking ability and navigation performance will be quantified

later with vehicle test data. Since the evaluation for the multipath alleviating ability needs

special experiments, it will be made in future. It is noted that the comparison results

above are still applicable to the comparison of the UT and TLA GPS/full IMU systems.

5.5 Test Description and Data Processing

To assess the performance of the UT GPS/RIMU systems and to verify the innovative

algorithms/configurations, the same data described in Section 3.3 was used. Data

176

processing was performed with the UT GPS/RIMU software developed from the TLA

GPS/RIMU software. To implement UT GPS/RIMU integration, the TLA GPS/RIMU

software was modified to include the vector tracking architectures, as discussed in

Sections 5.1 to 5.3. It is noted that the TLA GPS/RIMU software was originally

developed from the University of Calgary’s GSNRx™ software receiver (see Section 4.5).

In the data processing, two RIMU configurations (i.e. 3A1G and 2A1G) were used in the

assessment of the UT GPS/RIMU. To simplify analysis, only MEMS IMU data was used

in the assessment.

In UT system tests, the GPS pseudorange measurement noise variance (used to compute

the navigation solution) was chosen as 400 (m2) for open sky data, 900 (m

2) for foliage

data. The pseudorange rate measurement noise variance was chosen as 0.04 (m/s)2 for

both sets of data. For TLA system tests, the pseudorange measurement noise variance

was chosen as 100 (m2), and the pseudorange rate variance was chosen the same as that

of the UT system test, for both sets of test data. The reason for choosing larger

pseudorange noise variance for the UT system is that the pseudorange measurement of

the UT system comes from a VDLL, and the VDLL has more measurement thermal noise

than the DaDLL of the TLA system (see Section 5.1). Furthermore, since the C/N0 of

GPS signal of the foliage data varies dramatically (see Figure 4.14), its pseudorange

noise variance is chosen as larger than that of the open sky data. In contrast, to facilitate

comparison, the pseudorange rate variance for all tests was chosen as the same. For

Doppler-aided PLL, the noise bandwidth was chosen as 3 Hz for 3A1G, and 8 Hz for

2A1G. These were the best values that could be obtained through a series of noise

177

bandwidth selection trials. For CaPLL, the loop filter is only reset in the foliage test and

2A1G configuration. In other cases, the loop filter of the CaPLL is not reset, resulting in

a DaPLL. In the tests, the output rate of the integration system was 10 Hz, and PIT was

20 ms.

5.6 Evaluation of UT GPS/RIMU

In the following, the UT GPS/RIMU systems will be evaluated in terms of tracking

ability and navigation performance. The tracking ability of the GPS receiver is assessed

using the number of SVs used in the navigation solution, as applied in Section 4.6. First

the UT systems of VDF and VDCaP are evaluated and compared. Then the innovative

algorithms such as CaPF and RTL are investigated.

5.6.1 Evaluation of GPS/RIMU with VDF

With the open sky data, the velocity and attitude error of the UT GPS/RIMU (Crista) with

VDF have similar plots as those in Figure 4.20 for 3A1G and in Figure 4.21 for 2A1G.

The number of satellites used in the navigation solution is seven for both 3A1G and

2A1G in all the time. That means all the SVs are being used. The statistical results of the

UT GPS/RIMU (Crista) with VDF are summarized in Table 5.1. This table shows that for

both RIMU configurations (i.e. 3A1G and 2A1G), the RMS velocity error in each

direction is less than 0.05 m/s; the pitch and roll error are about 0.6 degrees, respectively;

and the azimuth error is less than 2.2 degrees.


Open Sky Data



UT GPS/3A1G 0.65 0.56 1.68 0.04 0.05 0.03

UT GPS/2A1G 0.63 0.59 2.13 0.04 0.05 0.04

178

For the foliage data, the velocity and attitude error variations of the UT system with VDF

are shown in Figure 5.10 for 3A1G, and Figure 5.11 for 2A1G. From these two figures, it

can be seen that the 2A1G has more vertical velocity error, which is consistent with the

results in Chapters Three and Four. Figure 5.12 displays the number of SVs used in the

navigation solution. It shows that although there are nine SVs in view, sometimes only

eight SVs are used in the navigation solution for both 3A1G and 2A1G. The SV that is

not used in the navigation solution calculation is rejected by an innovation sequence test.

Table 5.2 summarizes the statistical results of the UT system. From this table, it can be

seen that for both RIMU configurations, the RMS velocity error in each direction is less

than 0.07 m/s; the RMS pitch error is less than 0.4 degrees, the RMS roll error is about

0.6 degrees; and the RMS azimuth error is about 3.0 degrees. Further, comparing Table

5.1 and Table 5.2, it can be seen that with foliage data, the UT system has more velocity

and azimuth error, as is expected. It is not appropriate to compare the pitch/roll error of

the two sets of data because the two test routes have different local terrain.

179


Foliage Data


Foliage Data

180


with VDF and Foliage Data


Foliage Data



UT GPS/3A1G 0.34 0.62 2.91 0.06 0.07 0.03

UT GPS/2A1G 0.39 0.60 3.04 0.06 0.07 0.05

5.6.2 Evaluation of GPS/RIMU with VDCaP

The UT system with VDCaP has the similar velocity and attitude error variations as those

in the VDF with the same RIMU configuration and test data. Thus in the following only

the number of SVs used in the navigation solution and the statistical results are shown.

With open sky data, for 3A1G case, all the SVs in view (seven) are used in the navigation

solution. For the 2A1G case, occasionally one SV is rejected from the navigation solution,

as shown in Figure 5.13 (the rejected SV is PRN 11, its signal has some sudden signal

attenuations, as discussed in Section 4.7.1). The RMS errors of the UT system with

VDCaP are summarized in Table 5.3. From this table, it can be seen that the velocity

error in each direction is less than 0.04 m/s for 3A1G, and less than 0.05 m/s for 2A1G;

181

the pitch and roll error are about 0.6 degrees for both RIMU configurations; and the

azimuth error is about 2.5 degrees for 3A1G and 1.8 degrees for 2A1G.


with VDCaP and Open Sky Data


Open Sky Data



UT GPS/3A1G 0.65 0.58 2.53 0.04 0.04 0.03

UT GPS/2A1G 0.65 0.58 1.78 0.04 0.05 0.04

With the foliage data, the number of SVs used in the navigation solution is shown in

Figure 5.14. It can be seen that with VDCaP, 3A1G configuration has more SVs used in

navigation solution than 2A1G does. Furthermore, in both 3A1G and 2A1G, a few of

SVs in view are rejected from the navigation solution. The RMS errors are summarized

in Table 5.4. From this table, it can be seen that the velocity error can reach as large as

0.09 m/s for 3A1G (in the north direction), and 0.11 m/s for 2A1G (also in the north

direction). The pitch error is less than 0.5 degrees, and the roll error less than 0.7 degrees

for both RIMU configurations. The azimuth error is about 2.70 degrees for the 3A1G

case and 3.50 degrees for the 2A1G case. Comparing the results of 3A1G and 2A1G, it

can be seen that 2A1G configuration has larger velocity and azimuth error. This can be

182

explained in two ways. First, the noise bandwidth of the Doppler-aided PLL in the UT

GPS/2A1G is wide (i.e. 8 Hz), resulting in more Doppler measurement noise. Second,

more SVs are rejected from the navigation solution in the 2A1G case. Since the 2A1G

case has more Doppler measurement noise and fewer satellites used in navigation

solution, its navigation performance is degraded.


with VDCaP and Foliage Data


Foliage Data



UT GPS/3A1G 0.32 0.64 2.65 0.07 0.09 0.03

UT GPS/2A1G 0.45 0.68 3.50 0.08 0.11 0.07

The analysis now compares the results of the VDCaP with those of the VDF shown in

Section 5.6.1. For open sky data, Figure 5.13 shows that the VDCaP has a similar number

of satellites used in the navigation solution as the VDF (the VDF always has seven SVs

used in the navigation solution for both 3A1G and 2A1G). Also, both UT systems have a

similar navigation performance for corresponding RIMU configurations, as shown in

Table 5.1 and Table 5.3. For the foliage data, the VDF has much more SVs used in

183

navigation solution, as shown in Figure 5.12 and Figure 5.14. Comparing Table 5.2 and

Table 5.4, it can be seen that the VDF has less velocity error than does the VDCaP for a

corresponding RIMU configuration. This latter result is because there are more SVs used

in navigation solution in the VDF and the Doppler measurement error of the VDCaP is

not necessarily less than that of the VDF. For example, with the foliage data and 2A1G

configuration, for PRN 28 (with high elevation angle), the RMS Doppler measurement

error of the VDF is 0.26 Hz, but the RMS error of the VDCaP is 0.33 Hz. So, in this case,

the CaPLL (still a kind of DaPLL) has more Doppler measurement error than does the

VFLL. The reason is that the Doppler measurement error not only contains the Doppler

thermal noise, but also contains some Doppler errors caused by the dynamic state and

C/N0 variations, especially for the CaPLL because it is a closed loop, as discussed in

Section 5.1.2.1. As a result, although the theoretical analysis in Section 5.1.2.2 suggests

that a DaPLL might have less Doppler measurement thermal noise than a VFLL, when

other Doppler errors caused by the dynamic state and C/N0 variations are considered, the

total Doppler error of the DaPLL might be more than that of the VFLL.

5.6.3 Evaluation of GPS/RIMU with VDCaPF

Although the loop filter reset criteria of the CaPF and the CaPLL discussed in Section

5.1.2.1 are different, the VDCaPF and the VDCaP have a similar tracking performance,

i.e. they have a similar number of SVs used in navigation solution for a corresponding

RIMU configuration and test data. Consequently, the plots of the number of SVs used in

the navigation solution of the VDCaPF are omitted herein. The statistical results of the

UT GPS/RIMU with VDCaPF are summarized in Table 5.5 for open sky data, Table 5.6

for foliage data. In order to verify the CaPF approach, the test results of the UT

184

GPS/RIMU with VDCaP (in Figure 5.8, the output is always switched to upper branch),

whose CaPLL has the same loop filter reset criteria as that of the CaPF, are summarized

in Table 5.7 for open sky data, Table 5.8 for foliage data. Table 5.9 shows the statistical

results of the UT GPS/RIMU with VDCaPF, ALFs and open sky data.


Open Sky Data



UT GPS/3A1G 0.65 0.58 2.70 0.04 0.04 0.03

UT GPS/2A1G 0.65 0.58 1.79 0.04 0.05 0.04


Foliage Data



UT GPS/3A1G 0.33 0.63 2.66 0.07 0.08 0.03

UT GPS/2A1G 0.43 0.65 3.16 0.08 0.11 0.06


Open Sky Data (with Same Loop Filter Reset Criteria as CaPF)



UT GPS/3A1G 0.65 0.58 2.54 0.04 0.04 0.03

UT GPS/2A1G 0.65 0.58 1.76 0.04 0.05 0.04


Foliage Data (with Same Loop Filter Reset Criteria as CaPF)



UT GPS/3A1G 0.32 0.62 2.60 0.07 0.08 0.03

UT GPS/2A1G 0.44 0.68 3.61 0.08 0.10 0.07

185


Open Sky Data When ALF Is Applied



UT GPS/3A1G 0.65 0.57 2.44 0.04 0.04 0.03

UT GPS/2A1G 0.64 0.59 2.38 0.04 0.04 0.04

Comparing Table 5.5 and Table 5.7, it can be seen that for the open sky data, the

VDCaPF and the VDCaP have a similar navigation performance, although slightly

different azimuth errors. For the foliage data, comparing Table 5.6 and Table 5.8, similar

conclusions can be obtained as that of the open sky case. Therefore, it can be concluded

that the VDCaPF (or CaPF) approach is practicable although it cannot improve the

performance of the UT system obviously compared to the VDCaP (or just CaPLL) with

the same loop filter reset criteria. But as a different or an innovative configuration of the

carrier tracing loop, it is still attractive, and might potentially improve the performance of

the UT system in other environments. Further investigation is needed to explore its

potential advantages.

The reason the CaPF algorithm (or VDCaPF system) cannot improve the performance of

the UT system obviously compared to the VDCaP can be explained as follows.

According to the CaPF algorithm, only when the CaPLL is considered unlocked

(i.e. 0.9PLI < ) does the Doppler measurement come from the frequency discriminator.

But when 0.9PLI < , most of Doppler measurements are not used in the navigation

solution. Moreover, even if some Doppler measurements obtained from the frequency

discriminator are used in the navigation solution, the Doppler measurement difference

186

between the CaPLL and the frequency discriminator of the CaPF (calculated with

Equation (5.18)) may not be obvious. In this light, the navigation performance

improvement of the CaPF is not obvious. Actually in the above tests, for both the

VDCaPF and VDCaP, the Doppler measurements used in the navigation solution are

controlled by the status of bit synchronization (approximately equivalent to

0.9PLI ≥ case). Since the CaPF can provide a reliable Doppler measurement in both

carrier phase-locked (i.e. 0.9PLI ≥ ) and carrier phase-unlocked ( 0.9PLI < ) cases, the

Doppler measurements of the CaPF in carrier phase-unlocked case also can be used in the

navigation solution. In this light, compared to the VDCaP, the VDCaPF can have more

Doppler measurements used in the navigation solution, resulting in improved navigation

performance. That is why it is assumed that the VDCaPF can potentially improve the

navigation performance. However more tests are needed in the future to explore this

potential ablity of the VDCaPF.

Furthermore, comparing Table 5.3 to Table 5.7, and Table 5.4 to Table 5.8, it can be seen

that for the two different loop filter reset criteria (i.e. one is designed in Section 5.1.2.1,

the other is designed in Section 5.2 and applied in this section), the UT systems of

GPS/RIMU with VDCaP have slightly different performance. Specifically, for the open

sky data, the systems of VDCaP with the two different reset criteria have a similar

navigation performance; for foliage data, the navigation performance of the VDCaP with

the reset criteria of Section 5.2 is slightly better than that with the reset criteria of Section

5.1.2.1. This suggests that the loop filter reset criteria of Section 5.2 (used in this section)

is better than that of Section 5.1.2.1.

187

Further comparing Table 5.5 and Table 5.9, it can be seen that for the UT GPS/RIMU

with VDCaPF, the system with ALFs can achieve similar results with the system with

CNBLFs having a proper noise bandwidth. This means that the ALF algorithm, discussed

in Chapter Four, also can be used in the UT GPS/RIMU system.

5.6.4 Evaluation of GPS/RIMU with RTLs

In order to verify the innovative RTL algorithm, a partial GPS outage was simulated in

the data. In the open sky data, a 40 s long partial GPS outage was simulated by artificially

omitting some satellites from the navigation solution during post-mission processing,

resulting in only two satellites – PRNs 08 and 11 – to be included in the navigation

calculation. The partial outage lasted from GPS time 242290 s to 242330 s. During this

time, the GPS condition is not as favourable as in other periods, i.e. before and after the

simulated outage. Similarly, in the foliage data, the simulated partial GPS outage is 30 s

long, and only PRNs 08 and 28 were left in the navigation calculation during the outage.

The outage is from GPS time 238800 s to 238830 s. During this time, the number of SVs

used in the navigation solution in normal case is low (mostly about 3-4 SVs).

5.6.4.1 Test Results with Open Sky Data

The test results of open sky data are shown in Figure 5.15 and Figure 5.16. Specifically,

Figure 5.15 shows the number of SVs used in the navigation solution of UT GPS/3A1G

(upper figure) and the vector tracking status (lower figure). It can be seen that during the

partial GPS outage, only one or two satellites are used in the navigation solution. It is

noted that although there are two SVs which are left for the navigation calculation during

the partial outage, if the measurement error (i.e. pseudorange or Doppler shift error) of

188

one SV is too large, the SV will be rejected from the navigation solution calculation. This

rejection is implemented with a GPS measurement quality control algorithm (or

innovation sequence test) programmed in the navigation solution software. Accordingly,

if one SV is rejected in the navigation calculation, there is only one SV used in the

navigation solution during the partial outage. From Figure 5.15, it also can be seen that

after the partial GPS outage, the number of SVs used in the navigation solution will

recover to the normal case, i.e. without partial GPS outage case.


of UT GPS/3A1G (Crista) with RTLs and Open Sky Data

189


Open Sky Data

Figure 5.15 also shows the vector tracking status (lower figure), denoted as the indices of

CaPF and VDLL. If the index of CaPF is one, the carrier loop is a CaPF; if the index is

zero, the carrier loop is a PLL (without Doppler aiding, a third-order loop filter and a

noise bandwidth of 8 Hz). Similarly, if the index of VDLL is one, the code loop is a

VDLL; if the index is zero, the loop is a DaDLL (with code Doppler aiding and a narrow

noise bandwidth of 0.05 Hz). Finally, from Figure 5.15, it can be seen that when the

partial GPS outage occurred, the CaPF is the first to switch to the scalar tracking loop, i.e.

PLL. After the partial outage, the PLL and DLL switch back to the CaPF and VDLL,

respectively.

Figure 5.16 shows the velocity and attitude errors for the UT GPS/3A1G with RTLs.

From this figure, it can be seen that when the partial GPS outage occurs, both the velocity

and attitude error increase, and the increase of the velocity error is more noticeable. After

190

the partial outage, both the velocity and attitude error decrease, as should be expected.

Similar results are also obtained for the 2A1G case and are therefore not shown.

5.6.4.2 Test Results with Foliage Data

The test results of the foliage data are shown in Figure 5.17, Figure 5.18, and Figure 5.19.

Figure 5.17 shows the number of SVs used in the navigation solution of UT GPS/3A1G

and the vector tracking status. From this figure, it can be seen that when the partial GPS

outage occurred, the CaPF is the first to switch to the scalar tracking loop. Also, after the

partial GPS outage, the number of SVs used in the navigation solution recovers to the

normal case, and the PLL and DLL switch back to the CaPF and VDLL, respectively, just

like in the open sky case. Figure 5.18 shows the velocity and attitude error of the UT

GPS/3A1G with RTLs. It shows a similar result as in the open sky case, namely that

when the partial GPS outage occurs, both the velocity and attitude error increase, and,

that after the partial outage, both the velocity and attitude error decrease to normal values.

Figure 5.19 shows the number of SVs and the vector tracking status of the 2A1G

configuration. From this figure, it can be seen that there are two partial GPS outages in

the test: one is simulated and the other is naturally occurring. During the naturally

occurring partial outage, only the CaPF switches to the PLL. After either of the two

outages, the carrier loop and code loop switch back to the CaPF and VDLL, respectively.

After the simulated outage, the DLL is the first to switch back to the VDLL. From the

above results, it can be concluded that the RTL algorithm is valid, and can be used in

practice. Since, with the RTLs, the SVs in view can keep tracking during partial GPS

outages (and be used in the navigation solution), it is expected that the navigation

performance of the integration system during the partial GPS outages can be improved

191

compared to the UT system without RTLs. This cannot be confirmed because the

confirmation needs more tests.

Figure 5.17 Number of SVs Used in Navigation Solution and Vector Tracking

Status of UT GPS/3A1G (Crista) with RTLs and Foliage Data


Foliage Data

192


of UT GPS/2A1G (Crista) with RTLs and Foliage Data

5.6.4.3 Summary

From the above test results, it can be concluded that the RTL algorithm is valid. It can

switch a CaPF and a VDLL into a PLL and a DLL respectively when vector-based

tracking fails. Once the vector-based tracking is available, the tracking loops will switch

back to the CaPF and VDLL from the PLL and DLL, respectively. This will let the

satellites in view still keep tracking, thus saving the reacquisition step for those satellites,

and potentially improving the navigation performance of the GPS/RIMU, especially

during partial GPS outages.

5.6.5 Summary

Based on the test results presented above, the following conclusions are drawn:

• Both the GPS/RIMU with VDF and GPS/RIMU with VDCaP worked well.

193

• For the VDF, the velocity error in any one direction ranged between 0.03 and 0.07

m/s. Its pitch and roll error ranged between 0.34 and 0.65 degrees, respectively.

And the azimuth error ranged between 1.68 and 3.04 degrees.

• For the VDCaP (with the loop filter reset criteria of Section 5.1.2.1), its velocity

error ranged between 0.03 and 0.11 m/s. Its pitch and roll error ranged between

0.32 and 0.68 degrees, respectively. The azimuth error ranged between 1.78 and

3.50 degrees.

• In the open sky case, the VDF and VDCaP had a similar tracking ability (i.e. a

similar number of SVs used in navigation solution) and a similar navigation

performance in terms of velocity and attitude. In the foliage case, compared to the

VDF, the VDCaP performed worse, i.e. its tracking ability was low (i.e. the

number of SVs used in navigation solution was low), and its velocity error was

large.

• The CaPF is practicable. For the same loop filter reset criteria (designed in

Section 5.2), the VDCaPF and VDCaP have a similar navigation performance.

Their tracking abilities are similar because their carrier tracking loops are same,

i.e. both are CaPLL. Only their measurement outputs are different.

• The RTL algorithm is valid. It was shown to correctly switch a tracking loop

between a VBTL and a SBTL, thus saving the reacquisition for the satellites

which were in view during a partial GPS outage, and potentially improving the

navigation performance of the GPS/RIMU, especially during the partial GPS

outage.

194

It is noted that the above conclusions were obtained from the test data used in this

dissertation. Further testing would be required to confidently extend these conclusions to

other data sets. Further investigation is also needed to explore the potential advantages of

the CaPF approach.

5.7 Evaluation of UT and TLA GPS/RIMU

In the UT and TLA system comparison, the UT GPS/RIMU with VDCaP (designed in

Section 5.1.2.1) is chosen to compare with the corresponding TLA system, which is

implemented with a DaDLL plus a DaPLL (see Section 4.1). The multipath alleviating

comparison of the two systems is not made in this section because such an analysis would

need special experiments.

5.7.1 Test Results with Open Sky Data

With the open sky data, the tracking abilities of the UT and TLA GPS/RIMU are similar.

Specifically, the numbers of SVs used in the navigation solution are similar, i.e. seven

SVs are used for almost the entire test. The number of SVs used in the navigation

solution of the UT system is shown in Figure 5.13. The TLA system has a similar plot

(not shown to save the space). The test results of the UT and TLA GPS/RIMU are

summarized in Table 5.10. With reference to this table, “Azi.” is the abbreviation of

“Azimuth”, and “Nor.” is the abbreviation of “North”. From Table 5.10, it can be seen

that the UT and TLA systems have a similar navigation performance. Of particular

interest, however, is the azimuth error of the TLA GPS/2A1G which is noticeably greater

than that of the corresponding UT system.

195

Table 5.10 Position, Velocity, and Attitude Error of UT and TLA GPS/RIMU

(Crista) with Open Sky Data

RMS Attitude Error

(Deg)

RMS Velocity Error

(m/s)

RMS Position Error

(m) Integration

Strategy Pitch Roll Azi. East Nor. Up East Nor. Up

UT

GPS/3A1G 0.65 0.58 2.53 0.04 0.04 0.03 0.98 2.03 6.06

TLA

GPS/3A1G 0.65 0.58 2.54 0.04 0.04 0.03 0.83 1.69 6.27

UT

GPS/2A1G 0.65 0.58 1.78 0.04 0.05 0.04 0.96 2.01 6.32

TLA

GPS/2A1G 0.65 0.58 2.30 0.04 0.05 0.04 0.81 1.67 6.56

5.7.2 Test Results with Foliage Data

With the foliage data, the tracking performance of the UT and TLA GPS/RIMU are

shown in Figure 5.20 for 3A1G, and Figure 5.21 for 2A1G. From Figure 5.20, it can be

seen that the UT and TLA GPS/3A1G system have a similar number of SVs used in

navigation solution, i.e. they have a similar tracking ability. From Figure 5.21, it can be

seen that the TLA GPS/2A1G has more SVs used in navigation solution than does the

corresponding UT system. The UT GPS/2A1G loses one SV after the first few minutes

since the maximum number of SVs used in the navigation solution is only eight after the

first few minutes. Actually the SV (i.e. PRN 25, with low elevation angle, see Figure 4.13

– satellite skyplot), which is always not used in the navigation solution after the first few

minutes, does not lose tracking. It is just rejected from the navigation solution calculation

by a GPS measurement quality control algorithm programmed in the navigation solution

software, as discussed in Section 5.6.4. This rejection happens because the measurement

error (pseudorange/Doppler shift) of the SV is too large to be used in the navigation

solution. In other words, it is because of the measurement quality, not the loss of the SV’s

196

tracking that the SV is not used in the navigation solution. Actually since the rejected SV

(i.e. PRN 25) has a low elevation angle (about 23 degrees), its signal is weak, resulting in

a large pseudorange measurement error (about 300 km, the range of one C/A code). This

large error should be caused by the integer ambiguity in the code phase, resulting from

the problem in transmit time generation (e.g. in code decoding or bit synchronization). It

is noted that in the TLA GPS/2A1G, the carrier loop is a CaPLL, just like in the UT

system. So the difference between the UT and TLA GPS/2A1G is only in the code loop.


(Crista) with Foliage Data



The navigation performance of the UT and TLA GPS/RIMU is summarized in Table 5.11.

It can be seen that the UT and TLA system have a similar attitude error for either RIMU

197

configuration. For 3A1G configuration, the two systems have a similar velocity error, but

their position error is slightly different. This difference is mainly caused by the transient

process of the integration Kalman filter, as shown in Figure 5.22. From this figure, it can

be seen that the UT and TLA system have a similar position error in each direction after

the transient process. For 2A1G configuration, the TLA system has a smaller velocity

error (e.g. in north and vertical direction). Specifically, the 3D RMS velocity error of

TLA GPS/2A1G is reduced by 7.5% compared to the UT system. This could be caused

by the fact that the TLA GPS/2A1G has more SVs used in navigation solution, as

discussed above. The position error comparison of the two 2A1G systems has a similar

result as the 3A1G case. In short, the UT and TLA system have a similar position and

attitude error for either RIMU configuration. For 3A1G, the two systems also have a

similar velocity error. For 2A1G, the 3D RMS velocity error of the TLA system is

reduced by 7.5% compared to the UT system.

As discussed above, in the foliage test, the carrier loop of the TLA GPS/2A1G is

implemented with a CaPLL. In other cases, the carrier loop of the TLA system is

implemented with a DaPLL. In the TLA GPS/2A1G, in order to improve the system’s

performance in the foliage test, the PLL loop filter is reset according to a certain criterion,

as discussed in Section 5.1.2.1, resulting in a CaPLL. If the carrier loop of the TLA

system is implemented with a DaPLL, i.e. no loop filter reset for the DaPLL, the

navigation performance of the TLA GPS/2A1G is summarized in Table 5.12. Comparing

this table with the corresponding result in Table 5.11, it can be seen that with a CaPLL,

198

the TLA GPS/2A1G has a better navigation performance, especially in terms of velocity.

This further demonstrates that the CaPLL is valid.

Table 5.11 Position, Velocity, and Attitude Error of UT and TLA GPS/RIMU


RMS Attitude Error

(Deg)

RMS Velocity Error

(m/s)

RMS Position Error

(m) Integration


UT

GPS/3A1G 0.32 0.64 2.65 0.07 0.09 0.03 2.04 4.07 4.89

TLA

GPS/3A1G 0.33 0.63 2.71 0.07 0.08 0.03 4.18 1.57 7.08

UT

GPS/2A1G 0.45 0.68 3.50 0.08 0.11 0.07 1.96 4.32 4.97

TLA

GPS/2A1G 0.44 0.66 3.47 0.08 0.10 0.06 4.44 1.80 7.70

Table 5.12 Position, Velocity, and Attitude Error of TLA GPS/2A1G (Crista) With a

DaPLL and Foliage Data

RMS Attitude Error

(Deg)

RMS Velocity Error

(m/s)

RMS Position Error

(m) Integration


TLA

GPS/2A1G 0.48 0.70 3.41 0.09 0.12 0.06 4.30 2.46 7.33

Figure 5.22 Position Error of UT and TLA GPS/3A1G (Crista) with Foliage Data

199


Based on the above test result comparison, the following conclusions are drawn:

• The UT and TLA GPS/RIMU both have a similar tracking capability, i.e. a similar

number of SVs used in the navigation solution, except that the TLA GPS/2A1G

had more SVs used in navigation solution than the UT GPS/2A1G in the foliage

case (i.e. weak GPS signal case). The result that UT GPS/2A1G had less SVs used

in navigation solution in the foliage case should be caused by the code phase

integer ambiguity in the UT system test.

• The UT and TLA GPS/RIMU both have similar navigation performance in terms

of position, velocity, and attitude. The exception is that in foliage case, the 3D

RMS velocity error of TLA GPS/2A1G is reduced by 7.5% compared to the UT

system, resulting from the fact that the TLA GPS/2A1G had more SVs used in

navigation solution in the foliage case.

For the position error comparison of the UT and TLA system, the theoretical analysis of

Section 5.1.1.2 showed that a VDLL had larger pseudorange measurement thermal noise

than a DaDLL, which might result in a larger position error in a UT system. However, the

test results showed that the UT and TLA system had a similar position error. Thus the

theoretical and the test results seem to be somewhat inconsistent. Actually the position

error of the integration system might mainly come from the GPS measurement errors

such as multipath and other signal delays, not from the thermal noise. In this light,

although the VDLL has larger thermal noise, the UT and TLA system could still have a

similar position error.

200

5.8 Summary

In this chapter, first the UT GPS/RIMU was introduced, specifically for the UT

GPS/RIMU with VDF and UT GPS/RIMU with VDCaP. The configurations and

implementations of the two UT systems were discussed. Further, the measurement

thermal noises of the two systems were analyzed. Then two innovative

algorithms/configurations (i.e. CaPF and RTL) were developed. Among them, the CaPF

was used to improve the Doppler shift measurement, and the RTL was used to enhance

the system performance during partial GPS outages. Finally from the system

configuration and implementation, the UT system was compared with the corresponding

TLA system to identify their differences in both tracking and navigation performance.

Following the system design and analysis, system evaluations were conducted with the

vehicle test data. First the UT GPS/RIMU with VDF and UT GPS/RIMU with VDCaP

were evaluated in terms of tracking ability and navigation performance. Then the UT

GPS/RIMU with VDCaPF and UT GPS/RIMU with RTLs were assessed to verify these

two innovative algorithms, i.e. CaPF and RTL. Finally, a comparison of UT and TLA

systems was made with the test results of the two kinds of systems. In the above

evaluations, two RIMU configurations, i.e. 3A1G and 2A1G, were used. From the

evaluation results, the following conclusions are drawn:

• Both the GPS/RIMU with VDF and GPS/RIMU with VDCaP worked well. With

VDF, the velocity error in any one direction ranged between 0.03 and 0.07 m/s.

With VDCaP (designed in Section 5.1.2.1), the velocity error ranged between 0.03

201

and 0.11 m/s. The VDF and VDCaP had a similar attitude error for a

corresponding RIMU configuration and test data.

• With strong GPS signals (i.e. open sky case), VDF and VDCaP had a similar

tracking ability (i.e. a similar number of SVs used in navigation solution) and a

similar navigation performance. With weak signals (i.e. foliage case), VDF

outperformed VDCaP, especially in tracking ability and velocity accuracy.

• The innovative algorithm proposed herein – CaPF – is practicable. For the same

loop filter reset criteria (designed in Section 5.2), the VDCaPF and VDCaP have a

similar navigation performance (their tracking abilities are similar because their

carrier tracking loops are same, i.e. both are CaPLL).

• The innovative algorithm proposed herein – RTL – is valid. It correctly switched a

tracking loop between a VBTL and a SBTL, thus saving the reacquisition for the

satellites in view during a partial GPS outage, and potentially improving the

navigation performance of the GPS/RIMU during the partial GPS outage.

• The UT and TLA GPS/RIMU both had a similar tracking ability, except that the

TLA GPS/2A1G had more SVs used in navigation solution than the UT

GPS/2A1G in weak GPS signal case (i.e. foliage case). The result that UT

GPS/2A1G had less SVs used in navigation solution in the foliage case should be

caused by the code phase integer ambiguity in the UT system test.

• The UT and TLA GPS/RIMU both had a similar navigation performance, i.e. a

similar position, velocity, and attitude error, except that the 3D RMS velocity error

of TLA GPS/2A1G is reduced by 7.5% compared to the UT system in weak signal

202

case (i.e. foliage case), resulting from the fact that in weak signal case the TLA

GPS/2A1G had more SVs used in the navigation solution.

203

Chapter Six: Conclusions and Recommendations

This dissertation presented a thorough assessment for GPS/RIMU integration systems

used in land vehicle navigation. To be specific, the following approaches were

investigated in detail:

• An innovative LTP algorithm for RIMU error modeling in GPS/RIMU: In a

GPS/RIMU system, the pitch and roll cannot be calculated or observed directly

from the RIMU and the navigation performance is thus affected by local terrain

variations. To overcome this disadvantage, an LTP algorithm was developed for

the RIMU to improve the navigation performance, resulting in a set of innovative

mechanization equations and error model for RIMU. Furthermore, these equations

were compared with two other types of equations (namely DR and FD type) to

investigate their relative performance.

• An innovative ALF algorithm for TLA GPS/RIMU: In a TLA GPS/RIMU, since

the PLLs of the GPS receiver are aided with the Doppler shift, the noise

bandwidth of the loop filters of the PLLs can be narrowed more than in a GPS-

only case, resulting in an improved navigation performance. In order to take

advantage of this, an adaptive PLL loop filter whose noise bandwidth can be

adjusted according to the performance of the integrated system and GPS signal

C/N0 was developed to improve the performance of the TLA GPS/RIMU.

• Two innovative algorithms/configurations, namely CaPF and RTL, for UT

GPS/RIMU: In a UT GPS/RIMU, when the PLLs of the GPS receiver are not

locked, their Doppler measurements will be degraded by the transient state of the

PLLs. To overcome this disadvantage, a CaPF algorithm was developed to

204

provide more reliable Doppler measurement in both the “PLL locked” and “PLL

unlocked” cases. On the other hand, a reconfigurable tracking loop (i.e. RTL) was

developed to enhance the system performance during partial GPS outages. It can

switch the tracking loops of the GPS receiver between vector and scalar tracking

during partial GPS outages. In addition, two typical kinds of UT systems – VDF

and VDCaP – were implemented.

To assess the above innovative algorithms, the corresponding systems such as loose, TLA,

and UT GPS/RIMU were developed. Each of the above systems and innovative

algorithms was implemented in software (using the University of Calgary’s GSNRx™

software as a starting point, and then the corresponding software packages were

developed) and was evaluated with field vehicle test data. In the vehicle test, two

different grades of IMUs were used: a tactical-grade IMU (Honeywell HG1700) and a

MEMS-grade IMU (Crista). A GPS IF data collection system consisting of a NovAtel

Euro-3M card was used for IF data collection. Only two RIMU configurations suitable

for vehicle applications were considered: one is 3A1G, the other is 2A1G. A quite few

test runs were conducted to verify the above innovative algorithms and to assess the

different kinds of integrated systems (i.e. loose, TLA, and UT), thus providing useful

information for practical land vehicle navigation system development.

Through the above assessment for the loose, TLA, and UT GPS/RIM, a few of

conclusions were drawn. But only the major conclusions are outlined in the following

section.

205

6.1 Conclusions

Overall, the above innovative algorithms are valid, and the three types of GPS/RIMU,

namely loose, TLA, and UT, worked well in land vehicle navigation. The conclusions for

the different architectures are presented below.

Reduced IMU Mechanization (i.e. loose GPS/RIMU)

1. The LTP method can improve the navigation performance of a GPS/RIMU both

when GPS is available and unavailable. To be specific, with an LTP, pitch and roll

were estimated, and velocity error was reduced, especially in the horizontal

direction. The 3D RMS velocity error was reduced by more than 80% (from about

1.02 m/s to 0.14 m/s) compared to the case without LTP. During GPS outages, the

LTP method reduced both position and velocity errors.

2. For loose GPS/RIMU integration, the 2A1G may be a better configuration when

cost and performance are considered because with a vertical accelerometer, only

the vertical velocity error of the GPS/3A1G was reduced (e.g. for a MEMS IMU,

the vertical velocity error was reduced to 0.08 m/s from 0.09 m/s, about 11%).

3. The three types of M&E equations (i.e. LTP, DR, and FD model) are valid and can

be applied in the GPS/RIMU system for land vehicle navigation. Compared to the

LTP model, the DR model could achieve a similar performance only if the pitch

and roll were estimated in the accelerometer bias term (i.e., a composite bias);

otherwise its performance was much degraded. For the FD model, compared to the

LTP, it achieved a similar performance in strong GPS signal cases (i.e., open sky

case), but in weak signal cases (i.e., foliage case), its performance was inferior.

206

Specifically, the 3D RMS velocity error of the GPS/3A1G (Crista) with FD model

was more than twice that of the system with the LTP model.

TLA GPS/RIMU

1. In both ideal and real situations, namely on the roof of a building and on a real

road, the TLA GPS/RIMU performed better than the corresponding tight system in

terms of tracking ability and navigation solution accuracy. Specifically, with

Doppler aiding, the tracking performance of the GPS receiver was improved, with

more satellites being tracked, and smaller Doppler measurement errors. In addition,

with Doppler aiding, the navigation performance of the TLA GPS/RIMU was also

improved. More specifically, the 3D RMS velocity error was reduced by 54%

(from about 0.24 m/s to 0.11 m/s, for the foliage case), the pitch and roll error

were marginally reduced, if at all, but the azimuth error was reduced clearly.

Specifically in the foliage case, the RMS azimuth error of TLA GPS/3A1G (Crista)

was reduced by 26% (from about 3.64 degrees to 2.71 degrees) compared to the

tight GPS/3A1G (Crista).

2. The ALF algorithm can provide a proper noise bandwidth for the Doppler-aided

PLL of the TLA GPS/RIMU. As a result, compared to the CNBLF case, with

ALFs, the PLL tracking performance of the TLA GPS/RIMU was improved in

terms of reducing the Doppler measurement error and providing better phase

tracking, but some unmodeled system errors might cause phase lock degradation.

Furthermore, with ALFs, the navigation performance of the TLA GPS/RIMU was

improved, especially for velocity. The 3D RMS velocity error was reduced by up

207

to 19% (from about 0.17 m/s to 0.14 m/s). The pitch and roll error were also

reduced, but only slightly. The azimuth error was reduced by 17% (from about

3.28 degrees to 2.72 degrees) for MEMS IMU and the foliage test data.

3. In TLA GPS/RIMU, the 3A1G case outperformed the 2A1G case since it allowed

a narrower noise bandwidth, resulting in not only a smaller vertical velocity error

but also a smaller horizontal velocity error.

Ultra-Tight GPS/RIMU

1. Both the GPS/RIMU with VDF and GPS/RIMU with VDCaP worked well. With

VDF, the velocity error in any one direction ranged between 0.03 and 0.07 m/s.

With VDCaP (designed in Section 5.1.2.1), the velocity error ranged between 0.03

and 0.11 m/s. The VDF and VDCaP had a similar attitude error for a

corresponding RIMU configuration and test data.

2. With strong GPS signals, VDF and VDCaP had similar tracking ability and

navigation performance. With weak signals, VDF outperformed VDCaP,

especially in tracking ability and velocity accuracy.

3. For the CaPF, with the same loop filter reset criteria (designed in Section 5.2), the

VDCaPF and VDCaP have a similar navigation performance (their tracking

abilities are similar because their carrier tracking loops are same, i.e. both are

CaPLL). The RTL correctly switched a tracking loop between a VBTL and a

SBTL, thus saving the reacquisition for the satellites in view during a partial GPS

outage, and potentially improving the navigation performance of the GPS/RIMU

during the partial GPS outage.

208

4. UT and TLA GPS/RIMU both had similar tracking ability, except that the TLA

GPS/2A1G had more SVs used in the navigation solution than the UT GPS/2A1G

in a weak GPS signal case. The fact that UT GPS/2A1G had less SVs used in the

navigation solution in the foliage case was caused by the code phase integer

ambiguity in the UT system test. Thus, the tracking was fine, and the error could

conceivably be identified and corrected using innovation sequence in the

navigation solution, although this was beyond the scope of this work.

5. The UT and TLA GPS/RIMU both had similar navigation performance, except

that the velocity error of TLA GPS/2A1G is reduced by 7.5% (from about 0.15

m/s to 0.14 m/s) compared to the UT system. The fact that the TLA GPS/2A1G

had a smaller velocity error should result from the fact that in the weak GPS signal

case the TLA GPS/2A1G had more SVs used in navigation solution.

6.2 Recommendations

Based on the research and test results obtained, it can be seen that each method or system

has its own advantages and disadvantgaes. Thus, the best implementation choice depends

on the most important considerations for a particular system. For example, for RIMU

mechanization, compared to the LTP model, the DR model may not properly account for

all roll and pitch effects but it has a smaller compuation load and can be further

simplified. If the computation load and simplicity are important the DR model should be

selected. However generally the LTP model should be the best because it not only can

provide a similar or better performance in both strong and weak GPS signal cases, but it

also has a medium computation load compared to the DR and FD models. Similar rules

can be applied to the integration strategy choosing of the GPS/RIMU. Generally the

209

VDCaP or VDCaPF should be more preferred in land vehicle navigation because it not

only has the advantages of vector tracking loops (in code loop) compared to a scalar DLL,

but also has the advantages of PLL compared to an FLL.

For the future research, some recommendations can be made as follows.

Reduced IMU mechanization

1. Although the LTP algorithm is a valid attitude error model (for pitch and roll) for

reduced IMU, more experiments are needed to investigate the effect of the local

terrain and the GPS measurement accuracy on the navigation performance such as

attitude and velocity accuracy.

2. For MEMS reduced IMU, since the gyro measurement accuracy is low (its turn on

drift can be few thousands degrees per hour, see Section 3.3.1 for details), the

azimuth accuracy of the GPS/RIMU closely depends on the GPS measurement

accuracy and the vertical gyro’s drift estimation accuracy. Thus a proper error

model of the vertical gyro can benefit the performance of the GPS/RIMU. In this

light, the error model of the vertical gyro in a MEMS RIMU needs to be further

researched and properly modeled in the GPS/RIMU.

3. Furthermore, since in MEMS IMU, the azimuth accuracy of the GPS/RIMU

depends on the GPS measurement, when GPS measurement cannot give an

accurate enough azimuth estimation (i.e., within a few degrees), other on-vehicle

sensors such as a steering angle sensor should benefit the azimuth estimation,

especailly when the vehicle is moving slowly or is stopped. Moreover vehicle

210

speed sensors can mainly benefit the velocity estimation when GPS measurement

is poor. As a result, more research is needed to to explore the potential benfit of

the on-vehicle sensors for the proformance imporvement of the GPS/RIMU.

4. Although this dissertation conducted a thorough research on the RIMU

mechanization and error model, several questions still remain in order to develop a

practical GPS/RIMU system (i.e., a production system usable in all local terrain).

In particular, is it necessary that the local terrain model parameters be adjusted in

real time or for different local terrains and how to adjust the parameters if they

needs? Does the GPS measurement noise variance necessarily need to be adjusted

in real time? These topics should be further studied.

TLA GPS/RIMU

1. From the ALF results, it can be seen that some unmodeled system errors can cause

phase lock degradation during short time periods. Furthermore, different models

for Doppler aiding error can produce different noise bandwidth. In this light, the

ALF algorithm needs further research. In particular, a more precise Doppler error

model needs to be explored in the hope of further improving the performance of

the TLA system. In addition, the effect of the transient process of the tracking loop

and the C/N0 variations on the noise bandwidth choosing needs to be investigated.

2. In the TLA system, an OCXO was used in the GPS receiver. Such an oscillator has

a very low aging rate, high temperature stability, and low phase noise

(Symmetricom 2007). But its price is quite high (few thousands US dollars). For

land vehicle navigation system design, a low cost oscillator – TCXO – should be

211

used in practical TLA GPS/RIMU systems. Thus the performance of the TLA

system with a TCXO needs to be quantified.

Ultra-Tight GPS/RIMU

1. For different UT systems such as UT GPS/RIMU with VDF and UT GPS/RIMU

with VDCaP, they have different tracking and navigation performance. To further

quantify their differences in performance, more experiments need to be conducted,

such as experiments with different user dynamics and the experiments with

different C/N0 variations. Furthermore, the Doppler measurement errors of the

VFLL and the CaPLL need to be further researched and evaluated when user

dynamics and C/N0 variations are considered.

2. Further investigation is needed to explore the potential advantages of the CaPF

approach (i.e. the VDCaPF system).

3. From the above UT system research, it can be seen that in a UT system, the vector

tracking loop such as the VDLL only tracks the signals whose parameters (such as

time delay) can be known precisely enough. If the time delays of the tracked

signals are unknown or not known precisely enough, the VDLL will fail. As a

result, a VDLL not only has some advantages such as high anti-jamming and

multipath alleviating ability, but also has some disadvantages such as low stability,

compared to a DLL (a scalar tracking loop). In this light, innovative UT system

configurations/algorithms need to be developed to overcome the UT system’s

disadvantages. Furthermore, some special experiments such as urban canyon

212

experiments are needed to quantify the multipath alleviating ability of the UT

GPS/RIMU system.

213

References

Alban, S., D. Akos, S. Rock, and D. Gebre-Egziabher (2003) “Performance Analysis and

Architectures for INS-Aided GPS Tracking Loops,” in Proceedings of ION NTM

2003, 22-24 January, Anaheim CA, pp.611-622, U.S. Institute of Navigation,

Fairfax VA

Allen, J.M., J.H. Britt, C.J. Rose, and D.M. Bevly (2009) “Intelligent Multi-Sensor

Measurements to Enhance Vehicle Navigation and Safety Systems,” in

Proceedings of ION ITM 2009, 26-28 Jan., Anaheim, CA, pp. 74-83, U.S.

Institute of Navigation, Fairfax VA

Axelrad, P. and R.G. Brown (1996) “GPS Navigation Algorithms,” Chapter 9 of Global

Positioning System: Theory and Applications, Volume I, B.W. Parkinson and J.J.

Spilker Jr. (eds.), Washington, DC, American Institute of Aeronautics and

Astronautics, Inc., p. 409-433

Babu R.and J. Wang (2005) “Analysis of INS Derived Doppler Effects on Carrier

Tracking Loop,” The Journal of Navigation, vol 58, no 3, pp. 493-507, The Royal

Institute of Navigation

Benson, D. (2007) “Interference Benefits of a Vector Delay Lock Loop (VDLL) GPS

Receiver,” in Proceedings of ION 63rd Annual Meeting, 23-25 April, Cambridge,

Massachusetts, pp. 749-756, U.S. Institute of Navigation, Fairfax VA

Beser, J., A. Crane, S. Rounds, and J. Wyman (2002) “TRUNAVTM

: A Low-Cost

Guidance/Navigation Unit Integrating a SAASM-Based GPS and MEMS IMU in

a Deeply Coupled Mechanization,” in Proceedings of ION GPS 2002, 24-27

Sept., Portland, OR, pp. 545-555, U.S. Institute of Navigation, Fairfax VA

214

Best, R. (2007) Phase-Locked Loops: Design, Simulation, and Applications, Sixth

Edition, McGraw Hill

Borre, K., D.M. Akos, N. Bertelsen, P. Rinder, and S. H. Jensen (2007) A Software-

Defined GPS and Galileo Receiver: A Single-Frequency Approach, Birkhäuser

Britt, J.H. and D.M. Bevly (2009) “Lane Tracking using Multilayer Laser Scanner to

Enhance Vehicle Navigation and Safety Systems,” in Proceedings of ION ITM

2009, 26-28 Jan., Anaheim, CA, pp. 629-634, U.S. Institute of Navigation,

Fairfax VA

Brown, R.G. and P. Hwang (1992) Introduction to Random Signals and Applied Kalman

Filtering, John Wiley & Sons

Bullock, J.B., M. Foss, G.J. Geier, and M. King (2006) “Integration of GPS with Other

Sensors and Network Assistance,” Chapter 9 of Understanding GPS: Principle

and Applications, Second Edition, E.D. Kaplan and C.J. Hegarty (eds.), Norwood,

MA, Artech House, p.459-554

CCT (2006), Crista Inertial Measurement Unit (IMU) Interface / Operation Document,

Cloud Cap Technology Inc., USA, March (Available at

http://www.cloudcaptech.com)

Chiou, T (2005) “GPS Receiver Performance Using Inertial-Aided Carrier Tracking

Loop,” in Proceedings of ION GNSS 2005, 13-16 Sept., Long beach, CA,

pp.2895-2910, U.S. Institute of Navigation, Fairfax VA

Chiou, T., J. Seo, T. Walter, and P. Enge (2008) “Performance of a Doppler-Aided GPS

Navigation System for Aviation Applications under Ionospheric Scintillations,” in

215

Proceedings of ION GNSS 2008, 16-19 Sept., Savannah, GA, pp.1139-1147, U.S.


Chiou, T., D. Gebre-Egziabher, T. Walter, and P. Enge (2007) “Model Analysis on the

Performance for an Inertial Aided FLL-Assisted PLL Carrier-Tracking Loop in

the Presence of Ionospheric Scintillation, ” in Proceedings of ION NTM 2007, 22-

24 Jan., San Diego, CA, pp. 1276-1295, U.S. Institute of Navigation, Fairfax VA

Chiou, T., S. Alban, S. Atwater, J.D. Gautier, S. Pullen, P. Enge, D. Akos, D. Gebre-

Egziabher, and B. Pervan (2004) “Performance Analysis and Experimental

Validation of a Doppler-Aided GPS/INS Receiver for JPALS Applications,” in

Proceedings of ION GNSS 2004, 21-24 Sept., Long Beach, CA, pp. 1609-1618,

U.S. Institute of Navigation, Fairfax VA

Crane, R.N. (2007) “A Simplified Method for Deep Coupling of GPS and Inertial Data,”

in Proceedings of ION NTM 2007, 22-24 Jan., San Diego, CA, pp. 311-319, U.S.


Daum, P., J. Beyer, T. Köhler (1994) “Aided Inertial LAnd NAvigation System (ILANA)

with a Minimum Set of Inertial Sensors,” in Proceedings of IEEE-IEE Vehicle

Navigation & Information Systems Conference, Yokohama-VNIS’94, pp. 284-291

El-Sheimy, N. (2007) Inertial Techniques and INS/DGPS Integration, ENGO699.64

Course Notes, Department of Geomatics Engineering, University of Calgary,

Canada

Farrell, J. and M. Barth (1999) The Global Positioning System & Inertial Navigation,

McGraw-Hill

216

Gao, G. (2007a) INS-Assisted High Sensitivity GPS Receivers for Degraded Signal

Navigation, Ph.D. Dissertation, Department of Geomatics Engineering, University

of Calgary, Canada (Available at http://plan.geomatics.ucalgary.ca)

Gao, J. (2007b) Development of a Precise GPS/INS/On-Board Vehicle Sensors Integrated

Vehicular Positioning System, Ph.D. Dissertation, Department of Geomatics

Engineering, University of Calgary, Canada (Available at

http://plan.geomatics.ucalgary.ca)

Gao, J., M. Petovello and M.E. Cannon (2006) “Development of Precise GPS/INS/

Wheel Speed Sensor/Yaw Rate Sensor Integrated Vehicular Positioning system,”

in Proceedings of ION NTM 2006, 18-20 January, Monterey CA, pp.780-792,


Gautier, J.D. and B. Parkinson (2003) “Using the GPS/INS Generalized Evaluation Tool

(GIGET) for the Comparison of Loosely Coupled, Tightly Coupled and Ultra-

Tightly Coupled Integrated Navigation Systems,” in Proceedings of ION 59th AM

2003, 23-25 June, Albuquerque, NM, pp. 65-76, U.S. Institute of Navigation,

Fairfax VA

Gebre-Egziabher, D. (2007) “What is the difference between 'loose', 'tight', 'ultra-tight'

and 'deep' integration strategies for INS and GNSS ?,” GNSS Solutions Column,

Inside GNSS, vol 2, no 1, pp. 28-33

Gebre-Egziabher, D., A. Razavi, P. Enge, J.D. Gautier, S. Pullen, B. Pervan, and D. Akos

(2005) “ Sensitivity and Performance Analysis of Doppler-Aided GPS Carrier-

Tracking Loops,” Navigation, vol 52, no 2, Summer, pp. 49-60, U.S. Institute of

Navigation, Fairfax VA

217

Gelb, A. (1974) Applied Optimal Estimation, The M. I. T. Press, Massachusetts Institute

of Technology, Cambridge, Massachusetts, USA

Girod, B., R. Rabenstein, and A. Stenger (2001) Signals and Systems, John Wiley & sons

Godha, S. (2006) Performance Evaluation of Low Cost MEMS-Based IMU Integrated

with GPS for Land Vehicle Navigation Application, MSc Thesis, Department of

Geomatics Engineering, University of Calgary, Canada (Available at


Greenspan, R.L. (1996) “GPS and Inertial Integration,” Chapter 7 of Global Positioning

System: Theory and Applications, Volume II, B.W. Parkinson and J.J. Spilker Jr.

(eds.), Washington, DC, American Institute of Aeronautics and Astronautics, Inc.,

p. 187-218

Grewal, M.S. and A.P. Andrews (2001) Kalman Filtering: Theory and Practice Using

MATLAB, Second Edition, John Wiley & Sons, Inc.

Grewal, M.S., L.R. Weill, and A.P. Andrews (2001) Global Positioning Systems, Inertial

Navigation, and Integration, John Wiley & Sons, Inc.

Gustafson, D., and J. Dowdle (2003) “Deeply Integrated Code Tracking: Comparative

Performance Analysis,” in Proceedings of ION GPS/GNSS 2003, 9-12 Sept.,

Portland, OR, pp. 2553-2561, U.S. Institute of Navigation, Fairfax VA

Hamm, C., W. Flenniken, D. Bevly, and D. Lawerence (2004) “Comparative

Performance Analysis of Aided Carrier Tracking Loop Algorithms in High

Noise/High Dynamic Environments,” in Proceedings of ION GNSS 2004, 21-24

Sept., Long beach, CA, pp.523-532, U.S. Institute of Navigation, Fairfax VA

218

Hofmann-Wellenhof, B., K. Legat, and M. Wieser (2003) Principles of Positioning and

Guidance, Springer-Verlag Wien, NewYork

Hsieh, M. and G. Sobelman (2005) “Clock and Data Recovery with Adaptive Loop Gain

for Spread Spectrum SerDes Applications,” in Proceedings of IEEE ISCAS 2005,

23-26, May, Kobe, Japan, pp. 4883-4886

Humphreys, T., M.L. Psiaki, P. Kintner, and B. Ledvina (2005) “GPS Carrier Tracking

Loop Performance in the Presence of Ionospheric Scintillations,” in Proceedings

of ION GNSS 2005, 13-16 Sept., Long Beach CA, pp. 156-167, U.S. Institute of


Im, S., J. Song, B. Lee, G. Jee, S. Han, J. Bae, and J. Kim (2007) “Anti-Jamming

Technique Performance Evaluation for GPS L1 C/A Software Receiver,” in

Proceedings of ION GNSS 2007, 25-28 Sept., Fort Worth, TX, pp. 2787-2796,


Iqbal, U., A. F. Okou, and A. Noureldin (2008) “An Integrated Reduced Inertial Sensor

System -RISS / GPS for Land Vehicle,” in Proceedings of IEEE/ION PLANS

2008, 6-8 May, Monterey, CA, pp. 1014-1021

Jwo, D. (2001) “GPS Receiver Performance Enhancement via Inertial Velocity Aiding,”

The Journal of Navigation, vol 54, no 1, Feb., pp. 105-117, The Royal Institute of

Navigation

Kaplan, E.D., J.L. Leva, D. Milbert, and M.S. Pavloff (2006) “Fundamentals of Satellite

Navigation,” Chapter 2 of Understanding GPS: Principle and Applications,

Second Edition, E.D. Kaplan and C.J. Hegarty (eds.), Norwood, MA, Artech

House, p.21-65

219

Kennedy, S. and J. Rossi (2008) “Performance of a Deeply Coupled Commercial Grade

GPS/INS System from KHV and NovAtel Inc.,” in Proceedings of IEEE/ION

PLANS 2008, 6-8 May, Monterey, CA, pp. 17-24

Kiesel, S., M.M. Held, G.F. Trommer (2007) “Realization of a Deeply Coupled GPS/INS

Navigation System Based on INS-VDLL Integration,” in Proceedings of ION

NTM 2007, 22-24 January, San Diego CA, pp.522-531, U.S. Institute of


Kim, J., D. Hwang, and S. Lee (2007) “Performance Evaluation of INS-Aided Tracking

Loop and Deeply Coupled GPS/INS Integration System in Jamming

Environment,” in Proceedings of ION 63rd

Annual Meeting, 23-25 April,

Cambridge, Massachusetts, pp. 742-748, U.S. Institute of Navigation, Fairfax VA

Kuchar, J. (2001) “Markov Model of Terrain for Evaluation of Ground Proximity

Warning System Thresholds,” Journal of Guidance, Control, and Dynamics, vol

24, no 3, May–June, pp.428-435

Lachapelle, G. (2006) Navstar GPS Theory And Applications, ENGO 625 Course Notes,

Department of Geomatics Engineering, University of Calgary, Canada

Landis, D., T. Thorvaldsen, B. Fink, P. Sherman, S. Holmes (2006) “A Deeply

Integration Estimator for Urban Ground Navigation” in Proceedings of IEEE/ION

PLANS 2006, 25-27 April, San Diego CA, pp. 927-932

Lashley, M. and D. Bevly (2009a) “Vector Delay/Frequency Lock Loop Implementation

and Analysis,” in Proceedings of ION ITM 2009, 26-28 Jan., Anaheim, CA, pp.

1073-1086, U.S. Institute of Navigation, Fairfax VA

220

Lashley, M. and D. Bevly (2009b) “What are vector tracking loops, and what are their

benefits and drawbacks?,” GNSS Solutions Column, Inside GNSS, vol 4, no 3, pp.

16-21

Lashley, M. and D. Bevly (2008a) “Comparison of Traditional Tracking Loops and

Vector Based Tracking Loops for Weak GPS Signals,” in Proceedings of ION

NTM 2008, 28-30 Jan., San Diego, CA, pp. 310-316, U.S. Institute of Navigation,

Fairfax VA

Lashley, M., D. Bevly, and J. Hung (2008b) “Impact of Carrier to Noise Power Density,

Platform Dynamics, and IMU Quality on Deeply Integrated Navigation,” in

Proceedings of IEEE/ION PLANS 2008, 6-8 May, Monterey, CA, pp. 9-16

Lee, K., T. Lee, and K. Song (2007) “A Study on the Tracking Loop Design for Weak

Signal in High Dynamic Environment,” in Proceedings of ION 63rd

Annual

Meeting, 23-25, April, Cambridge, Massachusetts, pp. 589-594, U.S. Institute of


Li, T., M.G. Petovello, G. Lachapelle, and C. Basnayake (2009) “Performance Evaluation

of Ultra-tight Integration of GPS/Vehicle Sensors for Land Vehicle Navigation,”

in Proceedings of ION GNSS 2009, 22-25 Sept., Savannah, GA, pp.1785-1796,


Mandal, M., and A. Asif (2007) Continuous and Discrete Time Signals and Systems,

Cambridge University Press, p.395-410

Mao, W., H. Tsao, and F. Chang (2004) “A New Fuzzy Bandwidth Carrier Recovery

System in GPS for Robust Phase Tracking,” IEEE Signal Processing Letters, vol

11, no 4, April, pp. 431-434

221

Mattern, N., R. Schubert, and G. Wanielik (2008) “Image Landmark Based Positioning in

Road Safety Applications Using High Accurate Maps,” in Proceedings of

IEEE/ION PLANS 2008, 6-8 May, Monterey, CA, pp 1008-1013

Misra, P. and P. Enge (2001) Global Positioning System: Signals, Measurements, and

Performance, Ganga-Jamuna Press

Mohamed, A.H. and K.P. Schwarz (1999) “Adaptive Kalman Filtering for INS/GPS,”

Journal of Geodesy, vol 73, pp 193-203

Nassar, S., X. Niu, P. Aggarwal, and N. El-Sheimy (2006) “INS/GPS Sensitivity

Analysis Using Different Kalman Filter Approaches,” in Proceedings of ION

NTM 2006, 18-20 January, Monterey CA, pp.993-1001, U.S. Institute of


Nise, N. (2000) Control Systems Engineering, Third Edition, John Wiley & Sons

Niu, X., S. Nassar and N. El-Sheimy (2007a) “An Accurate Land-Vehicle MEMS

IMU/GPS Navigation System Using 3D Auxiliary Velocity Updates,” Navigation,

vol 54, no 3, pp.177-188, U.S. Institute of Navigation, Fairfax VA

Niu, X., S. Nasser, C. Goodall, and N. El-Sheimy (2007b) “A Universal Approach for

Processing any MEMS Inertial Sensor Configuration for Land-Vehicle

Navigation,” The Journal of Navigation, vol 60, no 2, pp.233-245, The Royal

Institute of Navigation

NovAtel (2009), SPAN™ Technology for OEMV User Manual, NovAtel Inc., Canada,

April (Available at http://www.novatel.ca)

Ochieng, W., J.W. Polak, R.B. Noland, J.Y. Park, L. Zhao, D. Briggs, J. Gulliver, A.

Crookell, R. Evans, M. Walker, W. Randolph (2003) “Integration of GPS and

222

Dead Reckoning for Real-time Vehicle Performance and Emissions Monitoring,”

GPS Solutions, vol 6, no 4, March, pp. 229-241

Ogata, K. (1997) Modern Control Engineering, Third Edition, Prentice-Hill

Ohlmeyer, E. J. (2006) “Analysis of an Ultra-Tight Coupled GPS/INS System in

Jamming,” in Proceedings of IEEE/ION PLANS 2006, 25-27 April, San Diego

CA, pp. 44-53

Pany, T., and B. Eissfeller (2006) “Use of a Vector Delay Lock Loop Receiver for GNSS

Signal Power Analysis in Bad Signal Conditions,” in Proceedings of IEEE/ION

PLANS 2006, 25-27 April, San Diego, CA, pp. 893-903

Pany, T., R. Kaniuth, and B. Eissfeller (2005) “Deep Integration of Navigation Solution

and Signal Processing,” in Proceedings of ION GNSS 2005, 13-16 Sept., Long

Beach, CA, pp. 1095-1102, U.S. Institute of Navigation, Fairfax VA

Petovello, M.G. (2003) Real-Time Integration of a Tactical-Grade IMU and GPS for

High-Accuracy Position and Navigation, Ph.D. Dissertation, Department of

Geomatics Engineering, University of Calgary, Canada (Available at


Petovello, M.G., C. O’Driscoll, and G. Lachapelle (2008a) “Carrier Phase Tracking of

Weak Signal Using Different Receiver Architectures,” in Proceedings of ION

NTM 2008, 28-30 Jan., San Diego, CA, pp. 781-791, U.S. Institute of Navigation,

Fairfax VA

Petovello, M.G., C. O’Driscoll, G. Lachapelle, D. Borio and H. Murtaza (2008b)

“Architecture and Benefits of an Advanced GNSS Software Receiver,” Journal of

Global Positioning System (JGPS), International Association of Chinese

223

Professionals in Global Positioning Systems, Vol 7, No 2, pp. 156-168. Available

at: http://www.gnss.com.au/JoGPS/v7n2/JoGPS_v7n2p156-168.pdf.

Petovello, M.G., D. Sun, G. Lachapelle and M.E. Cannon (2007) “Performance Analysis

of an Ultra-Tightly Integrated GPS and Reduced IMU System,” in Proceedings of

ION GNSS 2007, 25-28 Sept., Fort Worth, TX, pp. 602-609, U.S. Institute of


Petovello, M.G., and G. Lachapelle (2006a) “Comparison of Vector-Based Software

Receiver Implementations with Application to Ultra-Tight GPS/INS Integration,”

in Proceedings of ION GNSS 2006, 26-29 Sept., Fort Worth TX, pp.2977-2989,


Petovello, M.G., and G. Lachapelle (2006b) “An Efficient New Method of Doppler

Removal and Correlation with Application to Software-Based GNSS Receivers,”

in Proceedings of ION GNSS 2006, 26-29 Sept., Fort Worth TX, pp.2407-2417,


Phuyal, B. (2004) “An Experiment for a 2-D and 3-D GPS/INS Configuration for Land

Vehicle Applications,” in Proceedings of IEEE/ION PLANS 2004, 26-29 April,

Monterey CA, pp. 148-152

Psiaki, M.L. and H. Jung (2002) “Extended Kalman Filter Methods for Tracking Weak

GPS Signals,” in Proceedings of ION GPS 2002, 24-27 Sept., Portland, OR, pp.


Psiaki, M.L. (2001) “Smoother-Based GPS Signal Tracking in a Software Receiver,” in

Proceedings of ION GPS 2001, 11-14 Sept., Salt Lake City UT, pp. 2900-2913,


224

Rana, N.C. and P.S. Joay (2006) Classical Mechanics, Tata McGraw-Hill

Raquet, J.F. (2004) GPS Receiver Design, ENGO 699.10 Course Notes, Department of

Geomatics Engineering, University of Calgary, Canada

Rezaei, S. and R. Sengupta (2007) “Kalman Filter-Based Integration of DGPS and

Vehicle Sensors for Localization,” IEEE TRANSACTIONS ON CONTROL

SYSTEMS TECHNOLOGY, vol 15, no 6, Nov., pp. 1080-1088

Rogers, R.M. (1999) “Integrated DR/DGPS Using Low Cost Gyro and Speed Sensor,” in

Proceedings of ION NTM 1999, 25-27 Jan., San Diego, CA, pp. 353-360, U.S.


Sivananthan, S. and J. Weitzen (2009) “Improving Optimality of Deeply Coupled

Integration of GPS and INS,” in Proceedings of ION ITM 2009, 26-28 Jan.,

Anaheim, CA, pp. 426-433, U.S. Institute of Navigation, Fairfax VA

Soloviev, A. (2008) “Tight Coupling of GPS, Laser Scanner, and Inertial Measurements

for Navigation in Urban Environments,” in Proceedings of IEEE/ION PLANS


Soloviev, A., D. Bruckner, F. van Graas, and L. Marti (2007) “Assessment of GPS Signal

Quality in Urban Environments Using Deeply Integrated GPS/IMU,” in

Proceedings of ION NTM 2007, 22-24 January, San Diego, CA, pp. 815-828, U.S.


Soloviev, A., F. van Graas and S. Gunawardena (2004a) “Implementation of Deeply

Integrated GPS/Low-Cost IMU for Reacquisition and Tracking of Low CNR GPS

Signals,” in Proceedings of ION NTM 2004, 26-28 January, San Diego CA, pp.


225

Soloviev, A., S. Gunawardena and F. van Graas (2004b) “Deeply Integrated GPS/Low-

Cost IMU for Low CNR Signal Processing: Flight Test Results and Real Time

Implementation,” in Proceedings of ION GNSS 2004, 21-24 Sept., Long Beach

CA, pp.1598-1608, U.S. Institute of Navigation, Fairfax VA

Sönmez, T. and H.E. Bingöl (2008) “Modeling and Simulation of a Terrain Aided Inertial

Navigation Algorithm for Land Vehicles,” in Proceedings of IEEE/ION PLANS


Spilker Jr., J.J. (1996) “Fundamentals of Signal Tracking Theory,” Chapter 7 of Global

Positioning System: Theory and Applications, Volume I, B.W. Parkinson and J.J.

Spilker Jr. (eds.), Washington, DC, American Institute of Aeronautics and

Astronautics, Inc., p. 245-327

Stensby, J.L. (1997) Phase-Locked Loops: Theory and Applications, CRC Press

Sun, D., M.G. Petovello, and M.E. Cannon (2010) “Use of a Reduced IMU to Aid a GPS

Receiver with Adaptive Tracking Loops for Land Vehicle Navigation,” GPS

Solutions, doi:10.1007/s10291-009-0159-7

Sun, D., M.G. Petovello, and M.E. Cannon (2008) “GPS/Reduced IMU with a Local

Terrain Predictor in Land Vehicle Navigation,” International Journal of

Navigation and Observation, vol 2008, Article ID 813821, 15 pages, 2008.

doi:10.1155/2008/813821

Syed, Z., P. Aggarwal, X. Niu, and N. El-Sheimy (2007) “Economical and Robust

Inertial Sensor Configuration for a Portable Navigation System,” in Proceedings

of ION GNSS 2007, 25-28 Sept., Fort Worth TX, pp. 2129-2135, U.S. Institute of


226

Symmetricom (2007) “1000B Ultra-Stable Crystal Oscillator,” Symmetricom Inc, USA.

Available at: http://www.symmetricom.com

Titterton, D. and J. Weston (2004) Strapdown Inertial Navigation Technology, 2nd

Edition, The Institution of Electrical Engineers

Travis, W. and D.M. Bevly (2009) “Performance Analysis of a Closely Coupled

GPS/INS Relative Positioning Architecture for Automated Ground Vehicle

Convoys,” in Proceedings of ION ITM 2009, 26-28 Jan., Anaheim, CA, pp. 999-

1008, U.S. Institute of Navigation, Fairfax VA

Van Dierendonck, A.J. (1996) “GPS Receivers,” Chapter 8 of Global Positioning

System: Theory and Applications, Volume I, B.W. Parkinson and J.J. Spilker Jr.

(eds.), Washington, DC, American Institute of Aeronautics and Astronautics, Inc.,

p. 329-405

Van Dierendonck, A.J., P. Fenton, and T. Ford (1992) “Theory and Performance of

Narrow Correlator Spacing in a GPS Receiver,” Navigation, vol 39, no 3, Fall, pp.


Vlcek, C., P. McLain, M. Murphy (1993) “GPS/Dead Reckoning for Vehicle Tracking in

the ‘Urban Canyon’ Environment,” in Proceedings of IEEE-IEE Vehicle

Navigation & Information Systems Conference, Ottawa-VNIS’93, pp. A34-A41

Ward, P.W., J.W. Betz and C.J. Hegarty (2006) “Satellite Signal Acquisition, Tracking,

and Data Demodulation,” Chapter 5 of Understanding GPS: Principle and

Applications, Second Edition, E.D. Kaplan and C.J. Hegarty (eds.), Norwood,

MA, Artech House, p.153-240

227

Wendel, J., J. Metzger, R. Moenikes, A. Maier, and G.F. Trommer (2005) “A

Performance Comparison of Tightly Coupled GPS/INS Navigation Systems

Based on Extended and Sigma Point Kalman Filters,” in Proceedings of ION

GNSS 2005, 13-16 Sept., Long Beach CA, pp.456-466, U.S. Institute of


Xing, Z. and D. Gebre-Egziabher (2009) “Comparing Non-Linear Filters for Aided

Inertial Navigators,” in Proceedings of ION ITM 2009, 26-28 Jan., Anaheim, CA,

pp. 1048-1053, U.S. Institute of Navigation, Fairfax VA

Yu, W. (2007) Selected GPS Receiver Enhancements for Weak Signal Acquisition and

Tracking, MSc Thesis, Department of Geomatics Engineering, University of

Calgary, Canada (Available at http://plan.geomatics.ucalgary.ca)

Zhodzishsky, M., S. Yudanov, V. Veitsel, and J. Ashjaee (1998) “Co-Op Tracking for

Carrier Phase,” in Proceedings of ION GPS 1998, 15-18 Sept., Nashville

Tennessee, pp. 653-664, U.S. Institute of Navigation, Fairfax VA

Ziedan, N., J.L. Carrison (2003) “Bit Synchronization and Doppler Frequency Removal

at Very Low Carrier to Noise Ratio Using a Combination of the Viterbi

Algorithm with an Extended Kalman Filter,” in Proceedings of ION GPS/GNSS

2003, 9-12 Sept., Portland, OR, pp. 616-627, U.S. Institute of Navigation, Fairfax

VA

228

APPENDIX A: MECHANIZATION EQUATIONS AND ERROR MODEL OF

FULL IMU

The navigation equations and error model of a full IMU are expressed in local level

frame, and briefly introduced in the following.

A.1. Navigation Equations of Full IMU

The navigation equations of a full IMU are given by (El-Sheimy 2007; Titterton &

Weston 2004)

−=

++−=

= −

)(

)2(

1

b

i

b

ibbb

eie

b

b

ℓ

ℓℓ

ℓℓℓ

ℓ

ℓℓℓ

ℓℓ

ɺ

ɺ

ɺ

ΩΩRR

gvΩΩfRv

vDr

(A.1)

where [ ]×= ℓℓ

ieie ωΩ , [ ]×= ℓ

ℓ

ℓ

ℓ ee ωΩ , [ ]×= b

ib

b

ib ωΩ , [ ]×= b

i

b

i ℓℓωΩ , and a a

pq pq = × Ω ω are

skew-symmetric matrices, where a

pqω is a rotation rate column vector, from q-frame to p-

frame, and the vector is in a-frame. [ ]Tb

z

b

y

b

x

bfff=f is the measurement of

accelerometers, ( )Tb

ibz

b

iby

b

ibx

b

ib ωωω=ω is the measurement of gyros. [ ]Thλϕ=ℓr is

position, ϕ is latitude, λ is longitude, h is height. [ ]T

une vvv=ℓv is velocity, ev is

velocity in east, nv is velocity in north and uv is in up. 1−D is a matrix, which transfers

velocity in ℓ -frame into position variation rate in ℓ -frame. [ ]Tg−= 00ℓg , g is the

acceleration of gravity. ℓ

bR is a direction cosine matrix, from b -frame to ℓ -frame. For

more details,

229

0

cos

sin

ie e

e

ω ϕ

ω ϕ

=

ωℓ (A.2)

where eω is the rotation rate of the earth.

tan

n

ee

e

v

M h

v

N h

v

N h

ϕ

− + = +

+

ωℓ

ℓ (A.3)

Where M is meridian radius, N is prime vertical radius.

cos

tansin

n

b b b ei i e

ee

v

M h

v

N h

v

N h

ω φ

φω φ

− + = = + + +

+

ω R ω Rℓ

ℓ ℓ ℓ ℓ (A.4)

+

+

=−

100

00cos)(

1

01

0

1

ϕhN

hM

D (A.5)

A.2. Corresponding Error Model of Full IMU

The error model of a full IMU is given by (El-Sheimy 2007; Titterton & Weston 2004)

230

1 13 33 3 3 3 3 3

3 3

3 3

3 3 3 3 3 3 3 3

3 3 3 3 3 3 3 3

r

b fb

bi b

d

b

ω

δ δ

δ δ

− −×× × ×

×

×

× × × ×

× × × ×

−

− = + −

− −

0D D D 0 0 0r r

R WA B F 0 Rv v

R WP Q Ω R 0ε ε

W0 0 0 Γ 0d d

W0 0 0 0 Λb b

ℓ ℓ

ℓℓ ℓℓ ℓ

ℓℓ ℓℓ ℓ

ℓ

ɺ

ɺ

ɺ

ɺ

ɺ

(A.6)

where [ ]Thδδλδϕδ =ℓr is position error, [ ]T

une vvv δδδδ =ℓv is velocity error,

[ ]Tzyx

ℓℓℓℓ εεε=ε is attitude error in ℓ -frame, [ ]Tzyx ddd=d is gyro drift in b-

frame, [ ]Tzyx bbb=b is accelerometer bias in b-frame, [ ]×= ℓ

ℓ

ℓ

ℓ ii ωΩ . B is a matrix,

which transfers velocity error into velocity error varying rate in ℓ -frame. Q is a matrix

that transfers velocity error into attitude error rate in ℓ -frame. 1

r

−−D D , A and P are

matrices defined in El-Sheimy (2007). They give the terms related to position error in

position, velocity and attitude error equations. In order to simplify the error model

analysis, 1

r

−−D D , A and P are chosen as zero in this dissertation because the terms

related to these three matrices are small enough (Titterton & Weston 2004). [ ]×= ℓℓ fF ,

and the specific force in ℓ -frame is [ ]Tune fff=ℓf . )( zyxdiag ααα=Γ ,

)( zyxdiag βββ=Λ , 33×0 is a zero matrix with dimension 33× , 13×0 is a zero

vector with dimension 13× . dW and bW are driving noise. fW and ωW are the

accelerometer and gyro measurement white noise respectively. Other variables and

parameters are the same with those in Equation (A.1). For more details,

11 12 13

21 22 23

31 32 0

B B B

B B B

B B

=

B (A.7)

231

where hN

hMhB

+

++−=

)(tan11

ϕϕɺɺ

, ϕλω sin)2(12ɺ+= eB , ϕλω cos)2(13

ɺ+−= eB ,

ϕλω sin)(221ɺ+−= eB , )/(22 hMhB +−= ɺ , ϕɺ−=23B , ϕλω cos)(231

ɺ+= eB , ϕɺ232 =B .

+−

+−

+

=

00)/(tan

00)/(1

0)/(10

hN

hN

hM

ϕ

Q (A.8)

( )cos

( )sin

i e

e

ϕ

ω λ ϕ

ω λ ϕ

− = +

+

ωℓℓ

ɺ

ɺ

ɺ

(A.9)

232

APPENDIX B: INDIVIDUAL PHASE ERRORS OF DOPPLER-AIDED PLL

In the following, not only the formulae of thermal noise and oscillator noise-induced

phase jitters are given, but also the formulae of Doppler aiding error-induced phase errors

are derived. In the formula derivation, 707.0=ξ is the damping ratio of the second-order

system, nω is undamped natural frequency, T is Kalman filter measurement update

period.

B.1. Thermal Noise and Oscillator Noise-Induced Phase Jitters

PLL thermal noise jitter is given by (Ward et al 2006)

0 0

360 11

2 / 2 /

ntPLL

PIT

B

C N T C Nσ

π

= +

(degrees) (B.1)

where nB is carrier loop noise bandwidth (Hz), 0/ NC is carrier to noise power expressed

as a ratio (Hz), PITT is predetection integration time (seconds).

Allan deviation oscillator phase jitter for a second-order loop is given by (Chiou 2005)

nnn

A

hhh

ωω

π

ω

πσ

22

4180 2

2

3

3

4

2

+⋅

+⋅

= (degrees) (B.2)

where 2h , 3h , and 4h are coefficients for the oscillator, and given in Table B.1 (Chiou

2005; Gebre-Egziabher et al 2005).

233

Table B.1 Coefficients for Oscillator Error Models

kh TCXO OCXT

0h 85.00 10−× 85.50 10−×

1h 56.19 10−× 55.00 10−×

2h 49.60 10−× 46.50 10−×

3h 36.00 10−× 79.00 10−×

4h 46.00 10−× 71.00 10−×

Vibration-induced oscillator phase jitter is given by (Chiou 2005)

( ) mmvmev dffPfjH2

0)2(

180∫

∞

= ππ

σ (degrees) (B.3)

where )( mv fP is the PSD of phase jitter, which is defined in Ward et al (2006) and Chiou

(2005), and 2

2)(

)()(m

mg

mvmvf

fPfSfP = , where )( mv fS is oscillator vibration sensitivity,

)( mg fP is power curve of the random vibration, which is given in Table B.2 (Chiou 2005;

Gebre-Egziabher et al 2005), and mf is random vibration modulation frequency in Hz.

Since the parameters of random vibration given in Table B.2 was obtained from a

turbojet transport aircraft test, it should be bigger than that in a land vehicle navigation

test researched in this dissertation. The transfer function is

2

2 2

( )( )

( ) 2e

r n n

E s sH s

s s sδϕ ξω ω= =

+ +, where )(sE and ( )r sδϕ are defined in Figure 4.2.

Table B.2 Power Curve of Random Vibration (for aircraft)

3( ) 2.50 10g mP f−= × 40 Hzmf ≤

3 2( ) 2.50 10 ( / 40)g mP f f− −= × × 40 100 Hzmf< ≤

4( ) 4.00 10g mP f−= × 100 500 Hzmf< ≤

4 2( ) 4.00 10 ( / 500)g mP f f− −= × × 500 Hz mf<

234

B.2. Doppler Aiding Error-Induced Phase Errors

B.2.1. Random Ramp-Induced Phase Error

Random ramp-induced phase error can be obtained from the corresponding steady state

error, in response to the step of frequency derivative (see Figure 4.2). And the steady

state error is given by

2

2 2 20 0

( ( )) /lim ( ) lim ( ) lim ( )

2

dss d

t s sn n n

s d f t dte e t sE s F s

s sξω ω ω→∞ → →

− − ∆= = = ∆ =

+ + (B.4)

where 2 2

( ) ( )2

d

n n

sE s F s

s sξω ω

−= ∆

+ +.

B.2.2. First-Order GM-Induced Phase Jitter

First-order GM-induced phase jitter is derived as follows. From Equation (4.1), the PSD

of phase jitter can be expressed as

2( ) ( ) ( )fdP H j Pδϕ ω ω ω∆= (B.5)

where 22 2

)(nnss

ssH

ωξω ++= , ( )fdP ω∆ is the PSD of first-order GM Doppler error. In

sampling system, Doppler error Ttutfd <≤=∆ 0 ,)( 1 , ⋯ ,2 ,)( 2 TtTutfd <≤=∆ ,

iTtTiutf id <≤−=∆ )1( ,)( ,…, where iu is a constant in iTtTi <≤− )1( ,

Ni ..., ,3 ,2 ,1= . So the Fourier transform of )(tfd∆ can be expressed as

)(11

)(1

)1(

1)1(

TjTjN

i

Tij

i

TjN

i

iT

Ti

j

id eUj

eeu

j

edeujF

ωω

ωω

ωτ

ωωτω

−

=

−−−

=−

− −=

−==∆ ∑∑∫ (B.6)

From z-transform definition, when iu ( ⋯ ,3 ,2 ,1=i ) is a first-order GM sequence,

)( TjeU ω can be expressed as (Girod 2001)

235

TTj

Tj

w

Tj

ee

eeU

αω

ωω σ

−−=)( (B.7)

where α is the inverse of correlation time of the GM sequence, wσ is driving noise

sequence standard derivation. Since iu is sampling random signal, its variance is )(2tuδσ

(Mandal & Asif 2007). It can be approximated as Tu /2σ for its equivalent discrete signal

since ∫∫ ==TT

dtT

dtt00

11

)(δ . Therefore (Gelb 1974),

TeT

uw /)1( 2ασσ −−= (B.8)

where uσ is standard derivation of )(tu , where )(tu is the GM process. So the PSD of

)(tfd∆ can be expressed as

2 2

2 2

2

2 2

1 2 2cos( )( ) ( )

(1 ) 2 cos( )

k

fd d w T T

T

w

TP F j

e e T

e

α α

α

ωω ω σ

ω ωσ

ω γ

∆ − −

−= ∆ =

+ −⋅

≈+

(B.9)

where αγ α ⋅= Te . Therefore, the standard derivation of phase jitter is

2 22

4 4

1( )

2 22 2 2 2

T

w n

n n

eP d

α

δϕ δϕ

σ γ ω γσ ω ω

π ω γ ω

+∞

−∞

⋅= ≈ + −

+ ∫ (B.10)

If T andα are small enough, the following approximation can be made

22

32 2

w

n

δϕ

σσ

ω≈ (B.11)

In the derivation of Equation (B.10), the following formula is used (Chiou 2005)

( )∫∞+

−

=+0

1

/sin1 nmndx

x

xn

m

π

π, nm <<0 (B.12)

ultra-tight gps/reduced imu for land vehicle navigation · ultra-tight gps/reduced imu for land...

Documents